WEBVTT 00:00:00.000 --> 00:00:01.965 align:middle line:90% [SQUEAKING] 00:00:01.965 --> 00:00:02.465 align:middle line:90% 00:00:02.465 --> 00:00:03.937 align:middle line:90% [RUSTLING] 00:00:03.937 --> 00:00:04.437 align:middle line:90% 00:00:04.437 --> 00:00:07.888 align:middle line:90% [CLICKING] 00:00:07.888 --> 00:00:11.360 align:middle line:90% 00:00:11.360 --> 00:00:12.800 align:middle line:90% LAWRENCE GUTH: All right. 00:00:12.800 --> 00:00:16.200 align:middle line:84% Well, last class, we started to talk about the Fourier method 00:00:16.200 --> 00:00:18.440 align:middle line:90% in Euclidean space in R2. 00:00:18.440 --> 00:00:22.480 align:middle line:84% And the big goal for the class is to digest the Fourier method, 00:00:22.480 --> 00:00:26.040 align:middle line:84% finish that story, but also just to digest it in general. 00:00:26.040 --> 00:00:27.860 align:middle line:84% And then after that, if there's time, 00:00:27.860 --> 00:00:30.040 align:middle line:84% we'll start to introduce sieve theory, which 00:00:30.040 --> 00:00:32.270 align:middle line:90% has a extremely parallel story. 00:00:32.270 --> 00:00:34.520 align:middle line:84% But that was developed by a different group of people, 00:00:34.520 --> 00:00:36.070 align:middle line:84% and with somewhat different goals. 00:00:36.070 --> 00:00:38.960 align:middle line:90% 00:00:38.960 --> 00:00:40.940 align:middle line:84% OK, so to digest the Fourier method, 00:00:40.940 --> 00:00:43.520 align:middle line:84% I wanted to go back for a second to the finite field 00:00:43.520 --> 00:00:46.100 align:middle line:84% Fourier method, where everything is a little bit cleaner, 00:00:46.100 --> 00:00:47.740 align:middle line:90% and to make a point about it. 00:00:47.740 --> 00:00:51.000 align:middle line:90% 00:00:51.000 --> 00:01:04.209 align:middle line:84% OK, so our main lemma in the finite field case was this. 00:01:04.209 --> 00:01:12.730 align:middle line:84% If you have a set of lines in FQ2 00:01:12.730 --> 00:01:22.450 align:middle line:84% and then you add up their characteristic functions, 00:01:22.450 --> 00:01:26.050 align:middle line:84% then it has a big constant part, and then 00:01:26.050 --> 00:01:29.265 align:middle line:90% a small, high-frequency part. 00:01:29.265 --> 00:01:35.730 align:middle line:84% And then the conclusion is that f is f0 plus f high, 00:01:35.730 --> 00:01:39.690 align:middle line:84% where f0 is a constant function, which 00:01:39.690 --> 00:01:42.370 align:middle line:90% is the number of lines over q. 00:01:42.370 --> 00:01:46.730 align:middle line:84% And from that, you can compute its L2 norm. 00:01:46.730 --> 00:01:51.970 align:middle line:84% L2 squared is number of lines squared. 00:01:51.970 --> 00:01:56.010 align:middle line:90% And f high means 0. 00:01:56.010 --> 00:01:58.520 align:middle line:84% And we have a bound for its L2 norm. 00:01:58.520 --> 00:02:05.860 align:middle line:90% 00:02:05.860 --> 00:02:10.020 align:middle line:84% OK, so one thing that you can get out of this 00:02:10.020 --> 00:02:12.340 align:middle line:90% is a bound for the L2 norm of f. 00:02:12.340 --> 00:02:15.060 align:middle line:84% If you add these together, you get a bound for the L2 00:02:15.060 --> 00:02:17.140 align:middle line:90% norm of f. 00:02:17.140 --> 00:02:20.780 align:middle line:84% And it's worth saying that that bound for the L2 norm of f 00:02:20.780 --> 00:02:25.780 align:middle line:84% has a elementary proof that's really simpler than this. 00:02:25.780 --> 00:02:31.080 align:middle line:84% Let me show it for comparison to help put this in perspective. 00:02:31.080 --> 00:02:45.060 align:middle line:90% 00:02:45.060 --> 00:02:49.510 align:middle line:84% OK, so let's call this, elementary L2 bound. 00:02:49.510 --> 00:02:55.420 align:middle line:90% 00:02:55.420 --> 00:02:59.230 align:middle line:84% OK, so there's a first lemma that if you 00:02:59.230 --> 00:03:02.510 align:middle line:84% have two different lines, then you 00:03:02.510 --> 00:03:09.790 align:middle line:84% take the sum over x, L1 of x, L2 of x, this is at most 1. 00:03:09.790 --> 00:03:11.710 align:middle line:90% Could do that. 00:03:11.710 --> 00:03:16.030 align:middle line:84% It's just saying that two lines intersect in at most one point. 00:03:16.030 --> 00:03:19.330 align:middle line:84% OK, and we can use this to make an L2 estimate. 00:03:19.330 --> 00:03:20.670 align:middle line:90% So here's the L2 estimate. 00:03:20.670 --> 00:03:25.670 align:middle line:90% 00:03:25.670 --> 00:03:36.750 align:middle line:84% If f is the sum L in L, of L of x, then f L2 squared is bounded 00:03:36.750 --> 00:03:45.950 align:middle line:84% by the number of lines squared plus number of lines times q. 00:03:45.950 --> 00:03:48.590 align:middle line:90% Here's the proof. 00:03:48.590 --> 00:03:55.000 align:middle line:84% So f L2 squared is the sum over L1 and L2 00:03:55.000 --> 00:03:58.740 align:middle line:90% of sum on x, L1 of x, L2 of x. 00:03:58.740 --> 00:04:02.980 align:middle line:90% 00:04:02.980 --> 00:04:07.440 align:middle line:84% And now each one of these sums, we can control with this lemma. 00:04:07.440 --> 00:04:07.980 align:middle line:90% All right. 00:04:07.980 --> 00:04:09.647 align:middle line:84% But you have to be a little bit careful. 00:04:09.647 --> 00:04:12.300 align:middle line:84% If L1 is equal to L2, that's a different situation. 00:04:12.300 --> 00:04:14.640 align:middle line:84% A line intersects itself in a lot of points. 00:04:14.640 --> 00:04:17.279 align:middle line:84% So we're going to split up this sum. 00:04:17.279 --> 00:04:23.920 align:middle line:84% So it's the sum L1 equals L2 plus the sum L1 does not 00:04:23.920 --> 00:04:26.880 align:middle line:90% equal L2. 00:04:26.880 --> 00:04:33.080 align:middle line:84% OK, so here, we have only L choices for L1, 00:04:33.080 --> 00:04:35.920 align:middle line:84% or only L choices appear in this sum. 00:04:35.920 --> 00:04:38.690 align:middle line:84% And each one of them contributes q. 00:04:38.690 --> 00:04:41.680 align:middle line:90% 00:04:41.680 --> 00:04:46.300 align:middle line:84% And then over here, we have L squared choices for the lines. 00:04:46.300 --> 00:04:48.220 align:middle line:84% But each one of them contributes 1. 00:04:48.220 --> 00:04:51.880 align:middle line:90% 00:04:51.880 --> 00:04:55.050 align:middle line:90% So that's the proof. 00:04:55.050 --> 00:04:55.930 align:middle line:90% All right. 00:04:55.930 --> 00:04:59.205 align:middle line:84% So if you compare this with this, 00:04:59.205 --> 00:05:00.830 align:middle line:84% something looks kind of similar, right? 00:05:00.830 --> 00:05:04.330 align:middle line:84% We get an L2 bound up there of L squared plus Lq that 00:05:04.330 --> 00:05:07.050 align:middle line:84% matches L squared plus Lq that we get here with this quite 00:05:07.050 --> 00:05:09.730 align:middle line:90% simple argument. 00:05:09.730 --> 00:05:13.670 align:middle line:84% OK, but the main lemma has more information than that. 00:05:13.670 --> 00:05:16.290 align:middle line:90% 00:05:16.290 --> 00:05:19.810 align:middle line:84% Well, there's actually really two cases. 00:05:19.810 --> 00:05:23.770 align:middle line:84% If q is bigger than the number of lines, 00:05:23.770 --> 00:05:29.050 align:middle line:84% then this term dominates here, and this term dominates here. 00:05:29.050 --> 00:05:30.610 align:middle line:84% And in that situation, the main lemma 00:05:30.610 --> 00:05:33.330 align:middle line:84% doesn't really have much more information than this, 00:05:33.330 --> 00:05:36.690 align:middle line:84% because the function fh is not that special. 00:05:36.690 --> 00:05:39.050 align:middle line:84% But the other regime is that the number of lines 00:05:39.050 --> 00:05:40.350 align:middle line:90% is much bigger than q. 00:05:40.350 --> 00:05:43.610 align:middle line:90% So this dominates that. 00:05:43.610 --> 00:05:46.930 align:middle line:84% And in that situation, most of the L2 norm 00:05:46.930 --> 00:05:49.940 align:middle line:84% is coming from f0, which is very special because it's constant. 00:05:49.940 --> 00:05:55.120 align:middle line:90% 00:05:55.120 --> 00:05:57.900 align:middle line:84% So the main lemma has an important piece of information 00:05:57.900 --> 00:06:00.480 align:middle line:84% beyond this that when we have many lines, 00:06:00.480 --> 00:06:02.360 align:middle line:90% this sum is almost constant. 00:06:02.360 --> 00:06:07.260 align:middle line:84% And then it's a constant plus something that's much smaller. 00:06:07.260 --> 00:06:13.700 align:middle line:84% OK, so that is also true in spirit for the main lemma 00:06:13.700 --> 00:06:14.447 align:middle line:90% in the real case. 00:06:14.447 --> 00:06:16.780 align:middle line:84% But the main lemma in the real case is a little bit more 00:06:16.780 --> 00:06:17.440 align:middle line:90% complicated. 00:06:17.440 --> 00:06:19.300 align:middle line:84% So I wanted to do this first, and then we 00:06:19.300 --> 00:06:22.340 align:middle line:84% could look for the analogous thing in the more complicated 00:06:22.340 --> 00:06:24.460 align:middle line:90% real case. 00:06:24.460 --> 00:06:27.410 align:middle line:84% OK, so what did the main lemma say in the real case? 00:06:27.410 --> 00:06:35.660 align:middle line:90% 00:06:35.660 --> 00:06:39.740 align:middle line:90% Main lemma in the real case. 00:06:39.740 --> 00:06:43.700 align:middle line:84% So now, we have T is a set of 1 by R rectangles. 00:06:43.700 --> 00:06:47.940 align:middle line:90% 00:06:47.940 --> 00:06:55.086 align:middle line:84% And psi T is a smooth approximation 00:06:55.086 --> 00:06:58.110 align:middle line:84% of the characteristic function of t. 00:06:58.110 --> 00:07:00.430 align:middle line:84% And the exact meaning of that is explained 00:07:00.430 --> 00:07:05.390 align:middle line:84% in agonizing detail on the homework. 00:07:05.390 --> 00:07:07.390 align:middle line:84% OK, and then we have a function, which 00:07:07.390 --> 00:07:12.550 align:middle line:84% is the sum over all the T and T of this smooth characteristic 00:07:12.550 --> 00:07:15.150 align:middle line:90% function of T. 00:07:15.150 --> 00:07:18.630 align:middle line:84% OK, now our bounds are going to depend on how 00:07:18.630 --> 00:07:21.070 align:middle line:90% these tubes are clustered. 00:07:21.070 --> 00:07:28.070 align:middle line:84% So N sub T of R is the maximum over T tilde 00:07:28.070 --> 00:07:33.710 align:middle line:84% is a 2 little r by 2 big R rectangle, or tube, 00:07:33.710 --> 00:07:36.570 align:middle line:90% of the number of tubes in there. 00:07:36.570 --> 00:07:41.030 align:middle line:90% 00:07:41.030 --> 00:07:42.430 align:middle line:90% So picture this. 00:07:42.430 --> 00:07:45.090 align:middle line:90% We have this T tilde. 00:07:45.090 --> 00:07:49.450 align:middle line:84% Let me see how many little tubes are in T tilde. 00:07:49.450 --> 00:07:53.000 align:middle line:90% 00:07:53.000 --> 00:07:57.040 align:middle line:84% OK, so those are the hypotheses of the main lemma. 00:07:57.040 --> 00:08:01.040 align:middle line:84% And then the conclusion is that you can write f, 00:08:01.040 --> 00:08:05.400 align:middle line:84% you can break it up into pieces of different frequencies. 00:08:05.400 --> 00:08:08.080 align:middle line:84% So the pieces are labeled by a sort of a width parameter, 00:08:08.080 --> 00:08:11.080 align:middle line:84% little r, which is a dyadic number between scale 1 00:08:11.080 --> 00:08:16.960 align:middle line:84% and scale capital R of f sub little r, where the f sub 00:08:16.960 --> 00:08:19.020 align:middle line:84% little r has two good properties. 00:08:19.020 --> 00:08:21.320 align:middle line:90% It has a frequency support. 00:08:21.320 --> 00:08:24.360 align:middle line:84% The support of f little r hat is contained 00:08:24.360 --> 00:08:28.184 align:middle line:90% in the ball of radius 1 over r. 00:08:28.184 --> 00:08:32.640 align:middle line:90% And it has an L2 norm bound. 00:08:32.640 --> 00:08:41.000 align:middle line:84% f sub little r L2 squared is bounded by the number of tubes 00:08:41.000 --> 00:08:48.270 align:middle line:84% times this packing, clustering, parameter at the scale 00:08:48.270 --> 00:08:51.670 align:middle line:84% little r times big R over little r. 00:08:51.670 --> 00:08:57.870 align:middle line:90% 00:08:57.870 --> 00:08:58.370 align:middle line:90% OK. 00:08:58.370 --> 00:09:00.892 align:middle line:90% 00:09:00.892 --> 00:09:02.850 align:middle line:84% OK, now one thing you could take away from this 00:09:02.850 --> 00:09:05.130 align:middle line:84% is you could take away from it a bound for the L2 00:09:05.130 --> 00:09:08.330 align:middle line:84% norm of f by just adding up all of these. 00:09:08.330 --> 00:09:12.010 align:middle line:84% And like before, that bound for the L2 norm of f 00:09:12.010 --> 00:09:14.130 align:middle line:90% has a simpler proof. 00:09:14.130 --> 00:09:16.110 align:middle line:84% And for context for understanding this, 00:09:16.110 --> 00:09:19.350 align:middle line:84% I'm going to show you this simpler proof. 00:09:19.350 --> 00:09:19.850 align:middle line:90% Yes. 00:09:19.850 --> 00:09:23.290 align:middle line:84% AUDIENCE: Are the fr's orthogonal? 00:09:23.290 --> 00:09:27.090 align:middle line:84% LAWRENCE GUTH: The fr's are orthogonal. 00:09:27.090 --> 00:09:29.170 align:middle line:90% Yeah, the fr's are orthogonal. 00:09:29.170 --> 00:09:30.970 align:middle line:84% If we didn't know they were orthogonal, 00:09:30.970 --> 00:09:32.430 align:middle line:90% we would lose a little bit. 00:09:32.430 --> 00:09:36.795 align:middle line:84% We'd have to do Cauchy-Schwarz to compute the L2 norm of f. 00:09:36.795 --> 00:09:38.170 align:middle line:84% But the amount that we would lose 00:09:38.170 --> 00:09:42.070 align:middle line:84% would be like a log r, which is relatively minor. 00:09:42.070 --> 00:09:42.570 align:middle line:90% Yeah. 00:09:42.570 --> 00:09:46.060 align:middle line:90% 00:09:46.060 --> 00:09:46.560 align:middle line:90% OK. 00:09:46.560 --> 00:09:56.860 align:middle line:90% 00:09:56.860 --> 00:09:59.530 align:middle line:84% OK, so let's do an elementary L2 bound. 00:09:59.530 --> 00:10:07.900 align:middle line:90% 00:10:07.900 --> 00:10:08.700 align:middle line:90% All right. 00:10:08.700 --> 00:10:12.160 align:middle line:84% So like before, when we expand out the L2 norm, 00:10:12.160 --> 00:10:13.660 align:middle line:84% we'll have to estimate the integral 00:10:13.660 --> 00:10:16.620 align:middle line:90% of 1 tube times another tube. 00:10:16.620 --> 00:10:19.940 align:middle line:84% But one tube times another tube is a little bit more complicated 00:10:19.940 --> 00:10:22.080 align:middle line:84% than one line times another line. 00:10:22.080 --> 00:10:24.740 align:middle line:84% Two lines always intersect in a point, 00:10:24.740 --> 00:10:28.980 align:middle line:84% but two tubes intersect in a different amount of area 00:10:28.980 --> 00:10:31.260 align:middle line:84% depending upon the angle between them. 00:10:31.260 --> 00:10:34.700 align:middle line:84% So we're going to keep track of that. 00:10:34.700 --> 00:10:35.780 align:middle line:90% All right. 00:10:35.780 --> 00:10:38.660 align:middle line:90% So let's make a definition. 00:10:38.660 --> 00:10:44.030 align:middle line:84% The width, r, in between T1 and T2 00:10:44.030 --> 00:10:49.510 align:middle line:84% is the minimum r, so that there exists 00:10:49.510 --> 00:10:54.950 align:middle line:84% a 2 little r by 2 big R rectangle T tilde 00:10:54.950 --> 00:10:56.330 align:middle line:90% that contains them both. 00:10:56.330 --> 00:11:01.430 align:middle line:90% 00:11:01.430 --> 00:11:16.190 align:middle line:84% OK, so if this is T1, and this is T2, 00:11:16.190 --> 00:11:21.630 align:middle line:90% then that would be T tilde. 00:11:21.630 --> 00:11:22.930 align:middle line:90% And that would be r. 00:11:22.930 --> 00:11:25.950 align:middle line:90% 00:11:25.950 --> 00:11:36.350 align:middle line:84% OK, so now, if I want to integrate T1 of x, T2 of x, dx, 00:11:36.350 --> 00:11:41.070 align:middle line:84% that's going to be about big R over the little r associated 00:11:41.070 --> 00:11:42.960 align:middle line:90% to these two tubes. 00:11:42.960 --> 00:11:43.840 align:middle line:90% Why? 00:11:43.840 --> 00:11:47.680 align:middle line:84% Well, the integral is the area of their overlap. 00:11:47.680 --> 00:11:51.760 align:middle line:84% And if I'll sort of sketch their overlap over here-- 00:11:51.760 --> 00:11:53.680 align:middle line:90% it's not a great picture. 00:11:53.680 --> 00:11:57.200 align:middle line:84% This is around 1 because it's contained in one of the tubes. 00:11:57.200 --> 00:12:03.220 align:middle line:84% And this axis here is like big R over little r. 00:12:03.220 --> 00:12:05.820 align:middle line:84% As little r gets smaller, they overlap more and more. 00:12:05.820 --> 00:12:10.040 align:middle line:90% 00:12:10.040 --> 00:12:12.840 align:middle line:90% OK, good. 00:12:12.840 --> 00:12:14.730 align:middle line:90% So now, we can make an L2 bound. 00:12:14.730 --> 00:12:18.200 align:middle line:90% 00:12:18.200 --> 00:12:20.840 align:middle line:84% Let's say that the integral of f squared 00:12:20.840 --> 00:12:32.960 align:middle line:84% is bounded by the sum over little r dyadic of T times N sub 00:12:32.960 --> 00:12:38.240 align:middle line:84% T of little r times big R over little r. 00:12:38.240 --> 00:12:40.690 align:middle line:84% OK, and the proof is that we'll just 00:12:40.690 --> 00:12:45.330 align:middle line:84% break up this sum as a bunch of integrals like this, group 00:12:45.330 --> 00:12:47.010 align:middle line:84% them according to the value of little r, 00:12:47.010 --> 00:12:48.270 align:middle line:90% and we'll see what happens. 00:12:48.270 --> 00:12:50.570 align:middle line:84% This is what's going to fall out. 00:12:50.570 --> 00:12:53.530 align:middle line:90% So here's the proof. 00:12:53.530 --> 00:12:58.490 align:middle line:84% So the integral of f squared is the sum T1 and T2 00:12:58.490 --> 00:13:02.550 align:middle line:84% in our set of tubes integral of T1 of x, T2 of x. 00:13:02.550 --> 00:13:05.130 align:middle line:90% 00:13:05.130 --> 00:13:06.610 align:middle line:84% Now this integral here, it depends 00:13:06.610 --> 00:13:08.050 align:middle line:90% on the value of little r. 00:13:08.050 --> 00:13:12.210 align:middle line:84% So let's group these by the value of little r. 00:13:12.210 --> 00:13:18.970 align:middle line:84% So that's the sum over little r dyadic sum over T1 and T2 00:13:18.970 --> 00:13:21.970 align:middle line:90% with that value. 00:13:21.970 --> 00:13:26.930 align:middle line:84% So the R associated to T1 and T2 would be about this r. 00:13:26.930 --> 00:13:30.610 align:middle line:84% And then I have this integral, which I can bound by the lemma. 00:13:30.610 --> 00:13:32.670 align:middle line:90% So I get big R over little r. 00:13:32.670 --> 00:13:37.010 align:middle line:90% 00:13:37.010 --> 00:13:41.940 align:middle line:84% OK, now I need to know how many pairs of tubes are in this list 00:13:41.940 --> 00:13:43.180 align:middle line:90% here. 00:13:43.180 --> 00:13:44.900 align:middle line:90% Well, I can choose T1. 00:13:44.900 --> 00:13:47.000 align:middle line:84% And T1, as far as I know, could be anything. 00:13:47.000 --> 00:13:50.500 align:middle line:84% So I have number of tubes choices for T1. 00:13:50.500 --> 00:13:53.980 align:middle line:84% But once I choose T1, I need to choose another tube, T2, 00:13:53.980 --> 00:13:56.980 align:middle line:84% that fits into the same rectangle with T1. 00:13:56.980 --> 00:13:58.500 align:middle line:84% So the number of choices for that 00:13:58.500 --> 00:14:00.840 align:middle line:90% is about N sub T of little r. 00:14:00.840 --> 00:14:02.780 align:middle line:90% That's what it's measuring. 00:14:02.780 --> 00:14:13.180 align:middle line:84% So this is bounded by sum over r dyadic of number of tubes 00:14:13.180 --> 00:14:16.920 align:middle line:84% and N sub T of r big R over little r. 00:14:16.920 --> 00:14:20.620 align:middle line:90% 00:14:20.620 --> 00:14:22.820 align:middle line:90% OK, great. 00:14:22.820 --> 00:14:26.460 align:middle line:90% So let's compare this with this. 00:14:26.460 --> 00:14:29.340 align:middle line:84% This is exactly what we get by just adding up 00:14:29.340 --> 00:14:34.940 align:middle line:84% all those L2 norms for the f sub r. 00:14:34.940 --> 00:14:41.350 align:middle line:84% OK, so this proof is a lot easier 00:14:41.350 --> 00:14:45.150 align:middle line:84% than the proof of the main lemma that we did last time. 00:14:45.150 --> 00:14:47.890 align:middle line:84% Let's compare the L2 bound and the main lemma. 00:14:47.890 --> 00:14:53.590 align:middle line:90% 00:14:53.590 --> 00:14:56.990 align:middle line:84% If I were to erase this information in the main lemma, 00:14:56.990 --> 00:15:00.710 align:middle line:84% then all that would be left is the L2 bound. 00:15:00.710 --> 00:15:02.870 align:middle line:90% But there is this information. 00:15:02.870 --> 00:15:04.010 align:middle line:90% And this is helpful. 00:15:04.010 --> 00:15:07.030 align:middle line:84% This is telling us something more. 00:15:07.030 --> 00:15:10.370 align:middle line:84% OK, now depending upon what we know about this thing here, 00:15:10.370 --> 00:15:13.710 align:middle line:84% different terms might dominate this sum. 00:15:13.710 --> 00:15:17.910 align:middle line:84% If it so happens that the term little r equals 1, 00:15:17.910 --> 00:15:20.242 align:middle line:84% has the biggest contribution here, 00:15:20.242 --> 00:15:23.710 align:middle line:84% then we don't learn very much because our information 00:15:23.710 --> 00:15:27.630 align:middle line:84% about the support of f1 hat is the same as the information we 00:15:27.630 --> 00:15:29.710 align:middle line:84% already had about the support of f hat. 00:15:29.710 --> 00:15:31.750 align:middle line:90% So we'd have nothing new. 00:15:31.750 --> 00:15:35.475 align:middle line:84% But it may well happen that the term where this is biggest 00:15:35.475 --> 00:15:38.720 align:middle line:90% has little r much larger. 00:15:38.720 --> 00:15:43.860 align:middle line:84% And then f little r has this frequency support condition. 00:15:43.860 --> 00:15:46.520 align:middle line:84% It gives it some structure, some extra information there 00:15:46.520 --> 00:15:47.700 align:middle line:90% that's going to help us. 00:15:47.700 --> 00:15:50.920 align:middle line:90% 00:15:50.920 --> 00:15:53.720 align:middle line:84% So let's talk about, what does it 00:15:53.720 --> 00:15:56.600 align:middle line:84% mean about the function f little r that its frequency is 00:15:56.600 --> 00:15:57.460 align:middle line:90% supported there? 00:15:57.460 --> 00:16:21.040 align:middle line:90% 00:16:21.040 --> 00:16:26.240 align:middle line:84% So intuition, we're going to call 00:16:26.240 --> 00:16:32.836 align:middle line:84% this the locally-constant intuition, 00:16:32.836 --> 00:16:35.850 align:middle line:84% says that if I have some function g 00:16:35.850 --> 00:16:38.250 align:middle line:84% and the support of g hat is contained 00:16:38.250 --> 00:16:45.370 align:middle line:84% in the ball of radius 1 over r, then the function 00:16:45.370 --> 00:16:51.430 align:middle line:84% g is approximately constant on balls of radius r. 00:16:51.430 --> 00:17:01.679 align:middle line:90% 00:17:01.679 --> 00:17:08.530 align:middle line:84% So if the support of g hat is in the ball of radius 1 over r, 00:17:08.530 --> 00:17:13.210 align:middle line:84% then g hat is a combination of waves, cosine waves, 00:17:13.210 --> 00:17:17.714 align:middle line:84% where all of the cosine waves have frequency at most 1 over r. 00:17:17.714 --> 00:17:19.089 align:middle line:84% So each one of those cosine waves 00:17:19.089 --> 00:17:22.250 align:middle line:84% is not changing very much on balls of radius r. 00:17:22.250 --> 00:17:23.890 align:middle line:84% So it's sort of intuitively plausible 00:17:23.890 --> 00:17:25.730 align:middle line:84% that after we add them up, also, we'll 00:17:25.730 --> 00:17:28.790 align:middle line:84% get something that is roughly constant on balls of radius r. 00:17:28.790 --> 00:17:31.450 align:middle line:90% 00:17:31.450 --> 00:17:33.060 align:middle line:84% We'll come back in a little while 00:17:33.060 --> 00:17:35.040 align:middle line:84% and make a precise statement about this. 00:17:35.040 --> 00:17:37.620 align:middle line:90% 00:17:37.620 --> 00:17:42.340 align:middle line:84% OK, so based on this, let me make a picture illustrating 00:17:42.340 --> 00:17:43.360 align:middle line:90% main lemma r. 00:17:43.360 --> 00:17:48.580 align:middle line:90% 00:17:48.580 --> 00:17:51.570 align:middle line:90% Main lemma and the real case. 00:17:51.570 --> 00:17:54.940 align:middle line:90% 00:17:54.940 --> 00:18:03.740 align:middle line:90% OK, so OK. 00:18:03.740 --> 00:18:10.220 align:middle line:84% So this will be, well, this will be like each one of those lines 00:18:10.220 --> 00:18:12.260 align:middle line:90% is going to be a 1 by r tube. 00:18:12.260 --> 00:18:14.460 align:middle line:84% So I may have a whole bunch of them 00:18:14.460 --> 00:18:19.820 align:middle line:84% over here sort of like this, and a whole bunch of them over here 00:18:19.820 --> 00:18:20.650 align:middle line:90% sort of like this. 00:18:20.650 --> 00:18:24.780 align:middle line:90% 00:18:24.780 --> 00:18:27.560 align:middle line:84% And over here in this part of the board, 00:18:27.560 --> 00:18:32.073 align:middle line:84% I might have some that are not so clumped into thick tubes. 00:18:32.073 --> 00:18:33.490 align:middle line:84% They might look sort of like this. 00:18:33.490 --> 00:18:38.790 align:middle line:90% 00:18:38.790 --> 00:18:39.910 align:middle line:90% All right. 00:18:39.910 --> 00:18:43.270 align:middle line:84% So over here, a picture I wanted to give you 00:18:43.270 --> 00:18:46.790 align:middle line:84% is that I put so many thin strokes of chalk, one 00:18:46.790 --> 00:18:49.470 align:middle line:84% by our rectangles, that they sort of smooshed 00:18:49.470 --> 00:18:52.380 align:middle line:84% on top of each other, and they produced, like, a thick stroke. 00:18:52.380 --> 00:18:56.190 align:middle line:90% 00:18:56.190 --> 00:19:01.910 align:middle line:84% And so this part over here might be dominated by f of r. 00:19:01.910 --> 00:19:04.710 align:middle line:84% So maybe this scale might be r and f 00:19:04.710 --> 00:19:10.500 align:middle line:84% sub r be sort of a sum of this blob and that blob. 00:19:10.500 --> 00:19:14.110 align:middle line:90% 00:19:14.110 --> 00:19:19.670 align:middle line:84% And on the other, so this over here in blue would be f sub r. 00:19:19.670 --> 00:19:23.970 align:middle line:84% And on the other hand, this over here would be F sub 1. 00:19:23.970 --> 00:19:27.250 align:middle line:84% All of this stuff is pretty high frequency. 00:19:27.250 --> 00:19:33.760 align:middle line:90% 00:19:33.760 --> 00:19:37.960 align:middle line:84% OK, and notice that this f sub r is 00:19:37.960 --> 00:19:42.480 align:middle line:84% kind of roughly constant on any ball of radius r. 00:19:42.480 --> 00:19:55.400 align:middle line:84% And so if I were to visualize the set where F is big, 00:19:55.400 --> 00:19:59.510 align:middle line:84% well, there might be some bits in here where F is big. 00:19:59.510 --> 00:20:02.080 align:middle line:90% 00:20:02.080 --> 00:20:07.040 align:middle line:84% But then there also might be some piece in here 00:20:07.040 --> 00:20:07.730 align:middle line:90% where F is big. 00:20:07.730 --> 00:20:13.800 align:middle line:90% 00:20:13.800 --> 00:20:20.440 align:middle line:84% OK, so over here, this is where F sub 1 is big. 00:20:20.440 --> 00:20:23.790 align:middle line:84% And over here, this is where F sub r is big. 00:20:23.790 --> 00:20:29.490 align:middle line:90% 00:20:29.490 --> 00:20:30.330 align:middle line:90% All right. 00:20:30.330 --> 00:20:32.450 align:middle line:84% And now, the kind of trade off that happens 00:20:32.450 --> 00:20:37.370 align:middle line:84% is we might have a very strong bound for the L2 norm of f1. 00:20:37.370 --> 00:20:39.970 align:middle line:84% And so that bound tells me that this set where f1 is big 00:20:39.970 --> 00:20:40.640 align:middle line:90% is small. 00:20:40.640 --> 00:20:43.290 align:middle line:90% 00:20:43.290 --> 00:20:46.610 align:middle line:84% And then we have a bound for the L2 norm of fr. 00:20:46.610 --> 00:20:48.430 align:middle line:84% It might not be such a good bound. 00:20:48.430 --> 00:20:50.250 align:middle line:90% It might be bigger. 00:20:50.250 --> 00:20:53.050 align:middle line:84% But on the other hand, fr has this nice structure 00:20:53.050 --> 00:20:54.990 align:middle line:90% that it is constant on balls. 00:20:54.990 --> 00:20:58.370 align:middle line:84% So the set where fr is big is group 00:20:58.370 --> 00:21:00.810 align:middle line:90% is clumped into these big balls. 00:21:00.810 --> 00:21:02.490 align:middle line:90% And so that gives us-- 00:21:02.490 --> 00:21:05.370 align:middle line:84% So the area might be larger than this area. 00:21:05.370 --> 00:21:07.730 align:middle line:84% But the fact that it's clumped gives us 00:21:07.730 --> 00:21:09.615 align:middle line:84% some other geometric information about it. 00:21:09.615 --> 00:21:10.490 align:middle line:90% It's kind of simpler. 00:21:10.490 --> 00:21:17.130 align:middle line:90% 00:21:17.130 --> 00:21:19.790 align:middle line:90% OK, so let's pause there. 00:21:19.790 --> 00:21:24.330 align:middle line:84% That was recapping the main lemma in the real case. 00:21:24.330 --> 00:21:29.540 align:middle line:84% And I sort of tried to have a picture of what it's telling us. 00:21:29.540 --> 00:21:39.440 align:middle line:90% OK, OK. 00:21:39.440 --> 00:21:42.960 align:middle line:84% So the next thing that we should do is this is sort of nice, 00:21:42.960 --> 00:21:46.020 align:middle line:84% but it is not a actual precise math statement. 00:21:46.020 --> 00:21:48.960 align:middle line:84% So in order to really make use of this idea, 00:21:48.960 --> 00:21:51.380 align:middle line:84% we need to now replace this intuition 00:21:51.380 --> 00:21:54.900 align:middle line:90% by an actual precise thing. 00:21:54.900 --> 00:21:56.200 align:middle line:90% So let's do that. 00:21:56.200 --> 00:22:14.860 align:middle line:90% 00:22:14.860 --> 00:22:23.240 align:middle line:84% OK, so here's a precise version of the intuition. 00:22:23.240 --> 00:22:29.270 align:middle line:90% 00:22:29.270 --> 00:22:44.750 align:middle line:84% OK, OK, so how do we use the fact that the support of g hat 00:22:44.750 --> 00:22:47.410 align:middle line:84% is contained in something like B of 1 over r? 00:22:47.410 --> 00:22:50.870 align:middle line:90% 00:22:50.870 --> 00:22:57.830 align:middle line:84% There's a nice trick for using this, which goes like this. 00:22:57.830 --> 00:23:07.350 align:middle line:84% So suppose I have a function, eta so that eta of c 00:23:07.350 --> 00:23:12.830 align:middle line:84% is 1 for all the c in the ball of radius 1 over r. 00:23:12.830 --> 00:23:14.710 align:middle line:84% There are many such functions eta, 00:23:14.710 --> 00:23:17.210 align:middle line:84% and probably the most useful kind 00:23:17.210 --> 00:23:21.650 align:middle line:84% is to suppose that eta is also smooth and compactly supported. 00:23:21.650 --> 00:23:25.320 align:middle line:90% 00:23:25.320 --> 00:23:32.680 align:middle line:84% OK, so g hat is equal to g hat times eta, 00:23:32.680 --> 00:23:34.520 align:middle line:84% because g hat is supported on this ball, 00:23:34.520 --> 00:23:37.880 align:middle line:90% and eta is 1 on this ball. 00:23:37.880 --> 00:23:41.520 align:middle line:84% And now, I can take the inverse Fourier transform of this, 00:23:41.520 --> 00:23:45.980 align:middle line:84% and I get that g is g convolved with eta check. 00:23:45.980 --> 00:23:50.640 align:middle line:90% 00:23:50.640 --> 00:23:58.020 align:middle line:84% OK, so this follows from, and is almost equivalent to, this. 00:23:58.020 --> 00:24:02.400 align:middle line:84% But it's often a little bit easier to work with. 00:24:02.400 --> 00:24:04.640 align:middle line:84% So next to work with this, we should figure out 00:24:04.640 --> 00:24:07.630 align:middle line:84% a little bit what this function eta check is like. 00:24:07.630 --> 00:24:12.240 align:middle line:90% 00:24:12.240 --> 00:24:14.140 align:middle line:90% So properties of eta check. 00:24:14.140 --> 00:24:18.120 align:middle line:90% 00:24:18.120 --> 00:24:27.530 align:middle line:84% OK, so for one thing, eta check of x is less than r 00:24:27.530 --> 00:24:29.220 align:middle line:90% to the minus 2 in the plane. 00:24:29.220 --> 00:24:32.730 align:middle line:90% 00:24:32.730 --> 00:24:34.990 align:middle line:84% That just follows from the triangle inequality. 00:24:34.990 --> 00:24:36.930 align:middle line:84% To compute eta check, we integrate eta 00:24:36.930 --> 00:24:39.290 align:middle line:90% times the complex exponential. 00:24:39.290 --> 00:24:43.450 align:middle line:84% And eta has size 1, and it's supported on this ball. 00:24:43.450 --> 00:24:45.770 align:middle line:90% So we get this. 00:24:45.770 --> 00:24:50.650 align:middle line:84% But in addition to that, if x is large, then we get cancelation. 00:24:50.650 --> 00:24:57.490 align:middle line:84% So the next thing is that eta check of x decays rapidly. 00:24:57.490 --> 00:25:00.170 align:middle line:84% So I'll write something precise in a second. 00:25:00.170 --> 00:25:06.330 align:middle line:84% Decays rapidly if x is bigger than that. 00:25:06.330 --> 00:25:08.930 align:middle line:84% And so the reason for that is eta check 00:25:08.930 --> 00:25:16.610 align:middle line:84% of x is the integral eta of C e to the 2 pi i xc dc. 00:25:16.610 --> 00:25:21.740 align:middle line:84% And we can bound this integral by integrating by parts. 00:25:21.740 --> 00:25:24.600 align:middle line:84% And this is a nice and smooth thing. 00:25:24.600 --> 00:25:27.620 align:middle line:84% And if x is bigger than this, we integrate by parts many times, 00:25:27.620 --> 00:25:30.580 align:middle line:84% we get a strong estimate for this. 00:25:30.580 --> 00:25:33.140 align:middle line:84% OK, the estimate has the following shape. 00:25:33.140 --> 00:25:35.860 align:middle line:84% I start with an r to the minus 2 just 00:25:35.860 --> 00:25:37.860 align:middle line:90% to compare to what I had there. 00:25:37.860 --> 00:25:44.620 align:middle line:90% And then I have x over r. 00:25:44.620 --> 00:25:49.280 align:middle line:84% And I can put here any exponent I like. 00:25:49.280 --> 00:25:51.240 align:middle line:84% But negative 1,000 will be fine for us. 00:25:51.240 --> 00:25:54.460 align:middle line:90% 00:25:54.460 --> 00:25:57.700 align:middle line:84% So if x is more than a teeny bit bigger than r, 00:25:57.700 --> 00:25:59.000 align:middle line:90% this thing is very small. 00:25:59.000 --> 00:26:07.360 align:middle line:90% 00:26:07.360 --> 00:26:07.860 align:middle line:90% OK. 00:26:07.860 --> 00:26:38.870 align:middle line:90% 00:26:38.870 --> 00:26:41.670 align:middle line:90% All right. 00:26:41.670 --> 00:26:50.070 align:middle line:84% So now let me set C sub r to be the norm of eta check. 00:26:50.070 --> 00:26:53.990 align:middle line:84% And so C sub r has bounds that we just computed. 00:26:53.990 --> 00:26:59.430 align:middle line:84% So C sub r of x is bounded by r to the minus 2. 00:26:59.430 --> 00:27:07.010 align:middle line:84% And it's rapidly decaying if x is much bigger than r. 00:27:07.010 --> 00:27:09.630 align:middle line:90% 00:27:09.630 --> 00:27:12.270 align:middle line:90% And now, we can state a lemma. 00:27:12.270 --> 00:27:16.600 align:middle line:84% So if the support of g hat is contained 00:27:16.600 --> 00:27:22.960 align:middle line:84% in the ball of radius 1 over r, then the norm 00:27:22.960 --> 00:27:31.150 align:middle line:84% of g of x is bounded by the norm of g convolved with Cr x. 00:27:31.150 --> 00:27:36.680 align:middle line:90% 00:27:36.680 --> 00:27:41.800 align:middle line:84% And the proof is basically just what we have. 00:27:41.800 --> 00:27:45.480 align:middle line:84% So we started with g hat is g hat times eta. 00:27:45.480 --> 00:27:49.780 align:middle line:84% And so g is g convolved with eta check. 00:27:49.780 --> 00:27:54.680 align:middle line:90% 00:27:54.680 --> 00:27:57.080 align:middle line:84% Now take the triangle inequality. 00:27:57.080 --> 00:28:01.160 align:middle line:84% The norm of g is less than or equal to the norm of g convolved 00:28:01.160 --> 00:28:03.545 align:middle line:90% with the norm of eta check. 00:28:03.545 --> 00:28:04.420 align:middle line:90% And that's the proof. 00:28:04.420 --> 00:28:05.930 align:middle line:84% The norm of eta check is called Cr. 00:28:05.930 --> 00:28:18.690 align:middle line:90% 00:28:18.690 --> 00:28:21.170 align:middle line:84% OK, and since we're going to eventually 00:28:21.170 --> 00:28:23.770 align:middle line:84% be thinking about L2 norms, it is 00:28:23.770 --> 00:28:26.090 align:middle line:84% helpful to have a tiny variant of this lemma, 00:28:26.090 --> 00:28:27.850 align:middle line:84% where we have the norm of g squared 00:28:27.850 --> 00:28:29.688 align:middle line:90% instead of the norm of g. 00:28:29.688 --> 00:28:31.730 align:middle line:84% It's sort of convenient to prove it ahead of time 00:28:31.730 --> 00:28:34.490 align:middle line:84% instead of in the middle of what's coming. 00:28:34.490 --> 00:28:36.030 align:middle line:90% So lemma 2. 00:28:36.030 --> 00:28:38.970 align:middle line:90% 00:28:38.970 --> 00:28:41.970 align:middle line:84% If the support of g hat is contained 00:28:41.970 --> 00:28:47.730 align:middle line:84% in the ball of radius 1 over r, then the norm of g squared 00:28:47.730 --> 00:28:51.300 align:middle line:84% is bounded by the norm of g squared convolved with Cr. 00:28:51.300 --> 00:29:00.170 align:middle line:90% 00:29:00.170 --> 00:29:03.110 align:middle line:84% OK, so before we do the rigorous proof, 00:29:03.110 --> 00:29:06.730 align:middle line:84% let me say something about how I think about this. 00:29:06.730 --> 00:29:11.570 align:middle line:84% So this here says that the norm of g is bounded by-- 00:29:11.570 --> 00:29:13.683 align:middle line:84% this is sort of like taking an average. 00:29:13.683 --> 00:29:15.100 align:middle line:84% It's like the norm of g is bounded 00:29:15.100 --> 00:29:18.600 align:middle line:84% by the average value of the norm of G on a ball of radius r. 00:29:18.600 --> 00:29:22.730 align:middle line:90% 00:29:22.730 --> 00:29:26.080 align:middle line:84% And then if you had that by Cauchy-Schwarz, 00:29:26.080 --> 00:29:28.580 align:middle line:84% you would also have that the norm of g squared 00:29:28.580 --> 00:29:33.140 align:middle line:84% was bounded by the average of the norm of g 00:29:33.140 --> 00:29:35.940 align:middle line:90% squared on the ball. 00:29:35.940 --> 00:29:39.100 align:middle line:84% OK, so that's what we'll write down now. 00:29:39.100 --> 00:29:50.420 align:middle line:84% So the norm of g of x squared is bounded by the first lemma 00:29:50.420 --> 00:29:58.020 align:middle line:84% by the norm of g convolved with Cr at x squared. 00:29:58.020 --> 00:30:00.020 align:middle line:84% If I write out what this convolution is, 00:30:00.020 --> 00:30:09.590 align:middle line:84% it's the integral norm of g of x minus y, Cr of y dy all squared. 00:30:09.590 --> 00:30:12.110 align:middle line:90% 00:30:12.110 --> 00:30:14.710 align:middle line:84% And now, I can do a Cauchy-Schwarz. 00:30:14.710 --> 00:30:17.550 align:middle line:84% I want to do a Cauchy-Schwarz to get this guy squared 00:30:17.550 --> 00:30:19.350 align:middle line:90% with one of those. 00:30:19.350 --> 00:30:24.390 align:middle line:84% So when I do my Cauchy-Schwarz, my first factor 00:30:24.390 --> 00:30:29.730 align:middle line:84% will be integral norm of g squared x minus y Cr of y. 00:30:29.730 --> 00:30:32.550 align:middle line:90% 00:30:32.550 --> 00:30:37.280 align:middle line:84% And my second factor will just be the integral Cr of y dy. 00:30:37.280 --> 00:30:44.350 align:middle line:90% 00:30:44.350 --> 00:30:46.810 align:middle line:84% So you get to see that this is a valid Cauchy-Schwarz. 00:30:46.810 --> 00:30:49.230 align:middle line:84% The total number of exponents of g is 2. 00:30:49.230 --> 00:30:51.650 align:middle line:84% The total number of exponents of C is 2. 00:30:51.650 --> 00:30:55.150 align:middle line:84% And that matches what's on the other side. 00:30:55.150 --> 00:30:59.270 align:middle line:84% And now, this thing here is bounded 00:30:59.270 --> 00:31:04.070 align:middle line:84% by 1, which follows from our bounds for how it behaves, 00:31:04.070 --> 00:31:05.710 align:middle line:84% and which goes along with the intuition 00:31:05.710 --> 00:31:07.930 align:middle line:90% that this integral is 1. 00:31:07.930 --> 00:31:11.240 align:middle line:84% And so this convolution is like doing an average. 00:31:11.240 --> 00:31:13.720 align:middle line:84% OK, and this thing here is just the right-hand side. 00:31:13.720 --> 00:31:22.720 align:middle line:84% This is g squared convolved with C. OK, 00:31:22.720 --> 00:31:26.320 align:middle line:84% so this is the way we will rigorously make use of the fact 00:31:26.320 --> 00:31:29.140 align:middle line:90% that fr has frequency in there. 00:31:29.140 --> 00:31:32.960 align:middle line:90% 00:31:32.960 --> 00:31:35.160 align:middle line:84% And it corresponds visually to this picture 00:31:35.160 --> 00:31:41.440 align:middle line:84% that I think I'll erase now that fr looks of like this. 00:31:41.440 --> 00:31:44.800 align:middle line:84% And the places where it's big consists of some blobs of balls 00:31:44.800 --> 00:31:47.780 align:middle line:90% of radius r. 00:31:47.780 --> 00:31:48.280 align:middle line:90% OK. 00:31:48.280 --> 00:31:51.600 align:middle line:90% 00:31:51.600 --> 00:31:52.100 align:middle line:90% OK. 00:31:52.100 --> 00:31:56.720 align:middle line:90% 00:31:56.720 --> 00:31:58.840 align:middle line:84% So now, we can use our main lemma 00:31:58.840 --> 00:32:01.540 align:middle line:84% to prove some projection theory estimates. 00:32:01.540 --> 00:32:12.610 align:middle line:90% 00:32:12.610 --> 00:32:14.250 align:middle line:90% All right. 00:32:14.250 --> 00:32:19.250 align:middle line:84% So our setup is that we have x is 00:32:19.250 --> 00:32:25.170 align:middle line:84% a set of unit balls in the two-dimensional ball 00:32:25.170 --> 00:32:26.902 align:middle line:90% of radius r. 00:32:26.902 --> 00:32:28.630 align:middle line:90% D is the set of directions. 00:32:28.630 --> 00:32:33.210 align:middle line:84% So it's a subset of S1, which is 1 over r separated. 00:32:33.210 --> 00:32:37.510 align:middle line:84% And S measures the size of the biggest projection of x in here. 00:32:37.510 --> 00:32:41.850 align:middle line:84% So it's the maximum over all the thetas in my direction set 00:32:41.850 --> 00:32:43.870 align:middle line:90% of the theta projection of x. 00:32:43.870 --> 00:32:46.677 align:middle line:90% 00:32:46.677 --> 00:32:47.510 align:middle line:90% So that's our setup. 00:32:47.510 --> 00:32:50.850 align:middle line:84% And we're going to try to bound these different quantities 00:32:50.850 --> 00:32:52.850 align:middle line:84% in terms of how the directions and how 00:32:52.850 --> 00:32:54.610 align:middle line:90% the spheres are clustered. 00:32:54.610 --> 00:33:00.610 align:middle line:84% So remember, Nx of r is the maximum over all possible center 00:33:00.610 --> 00:33:06.610 align:middle line:84% C of the amount of x in the ball of center C and radius r. 00:33:06.610 --> 00:33:15.220 align:middle line:84% And similarly, N sub d of rho is the maximum over arcs 00:33:15.220 --> 00:33:23.817 align:middle line:84% and S1 of length rho of the number of directions in the arc. 00:33:23.817 --> 00:33:24.650 align:middle line:90% So that's our setup. 00:33:24.650 --> 00:33:27.620 align:middle line:90% 00:33:27.620 --> 00:33:30.260 align:middle line:84% And the projection theory estimate that comes from this 00:33:30.260 --> 00:33:34.300 align:middle line:84% method, I'll call it theorem 2R, theorem 2 in the real case, 00:33:34.300 --> 00:33:42.260 align:middle line:84% says that if we have this setup, then the number of directions is 00:33:42.260 --> 00:34:01.780 align:middle line:84% bounded by S big R over x times the maximum over little r of Nx 00:34:01.780 --> 00:34:07.080 align:middle line:84% of little r ND of little r over big R over little r squared. 00:34:07.080 --> 00:34:11.949 align:middle line:90% 00:34:11.949 --> 00:34:14.670 align:middle line:84% OK, so this is a little bit messy. 00:34:14.670 --> 00:34:16.710 align:middle line:84% And we'll eventually make some hypotheses 00:34:16.710 --> 00:34:19.830 align:middle line:90% that allow it to simplify it. 00:34:19.830 --> 00:34:22.909 align:middle line:84% But just notice for now that this estimate 00:34:22.909 --> 00:34:25.810 align:middle line:84% takes account of how the set of directions is clustered. 00:34:25.810 --> 00:34:27.489 align:middle line:84% And it takes account of how-- sorry. 00:34:27.489 --> 00:34:31.370 align:middle line:84% It takes account of how the set x of unit balls is clustered. 00:34:31.370 --> 00:34:34.050 align:middle line:84% And it takes account of how the set of directions is clustered. 00:34:34.050 --> 00:34:37.389 align:middle line:90% 00:34:37.389 --> 00:34:42.070 align:middle line:84% OK, so the proof follows the idea of the proof 00:34:42.070 --> 00:34:45.670 align:middle line:84% we did in finite fields, and makes use of the main lemma. 00:34:45.670 --> 00:34:47.989 align:middle line:90% And here is our setup. 00:34:47.989 --> 00:34:51.650 align:middle line:84% So for every direction in our direction set, 00:34:51.650 --> 00:34:55.989 align:middle line:84% let's say T theta, is a set of at most, 00:34:55.989 --> 00:35:03.130 align:middle line:84% S. We could say exactly S. 1 by r tubes T that cover x. 00:35:03.130 --> 00:35:07.320 align:middle line:90% 00:35:07.320 --> 00:35:09.480 align:middle line:84% So this projection has size S. Then 00:35:09.480 --> 00:35:11.600 align:middle line:84% looking at the fibers of the projections, 00:35:11.600 --> 00:35:16.480 align:middle line:84% I get S tubes that cover my set x. 00:35:16.480 --> 00:35:20.040 align:middle line:84% OK, then T, my master set of tubes, 00:35:20.040 --> 00:35:22.680 align:middle line:84% will be the union over all the directions. 00:35:22.680 --> 00:35:23.920 align:middle line:90% T theta. 00:35:23.920 --> 00:35:27.080 align:middle line:84% And f is going to be the sum over all 00:35:27.080 --> 00:35:32.720 align:middle line:84% these tubes of the smoothed characteristic function 00:35:32.720 --> 00:35:35.640 align:middle line:90% of the tube. 00:35:35.640 --> 00:35:39.920 align:middle line:84% OK, so we observe that for every x 00:35:39.920 --> 00:35:46.220 align:middle line:84% in x, f of x is at least the number of directions. 00:35:46.220 --> 00:35:55.880 align:middle line:90% 00:35:55.880 --> 00:35:57.840 align:middle line:90% All right. 00:35:57.840 --> 00:36:02.330 align:middle line:84% So now, we use our main lemma and we break up f into these 00:36:02.330 --> 00:36:05.165 align:middle line:90% different fr's. 00:36:05.165 --> 00:36:06.870 align:middle line:90% We use main lemma. 00:36:06.870 --> 00:36:10.850 align:middle line:90% 00:36:10.850 --> 00:36:13.530 align:middle line:84% And there are only logarithmically many different 00:36:13.530 --> 00:36:14.750 align:middle line:90% choices of r. 00:36:14.750 --> 00:36:17.730 align:middle line:84% And so one of them has to make a pretty big contribution 00:36:17.730 --> 00:36:19.370 align:middle line:90% at each of these points. 00:36:19.370 --> 00:36:25.170 align:middle line:90% And so I can choose r so that-- 00:36:25.170 --> 00:36:26.370 align:middle line:90% all right. 00:36:26.370 --> 00:36:29.210 align:middle line:90% So I have x times D squared. 00:36:29.210 --> 00:36:33.930 align:middle line:84% That's smaller than the integral over x of f squared. 00:36:33.930 --> 00:36:35.590 align:middle line:84% And now, there are only a few r's. 00:36:35.590 --> 00:36:38.650 align:middle line:84% So that should be somewhat smaller 00:36:38.650 --> 00:36:42.840 align:middle line:84% than a log times the integral over x of r squared. 00:36:42.840 --> 00:36:54.250 align:middle line:90% 00:36:54.250 --> 00:37:03.340 align:middle line:84% OK, now the first thing I might try here 00:37:03.340 --> 00:37:06.120 align:middle line:84% is if you look at the statement of the main lemma, 00:37:06.120 --> 00:37:08.780 align:middle line:84% it has an upper bound for the L2 norm of fr. 00:37:08.780 --> 00:37:13.177 align:middle line:84% And I could plug in that upper bound, and I would bound this. 00:37:13.177 --> 00:37:14.760 align:middle line:84% And I would get something out of that. 00:37:14.760 --> 00:37:18.580 align:middle line:90% 00:37:18.580 --> 00:37:23.257 align:middle line:84% But that's not the optimal thing to do here. 00:37:23.257 --> 00:37:25.340 align:middle line:84% Well, depending on what we know about x, depending 00:37:25.340 --> 00:37:27.213 align:middle line:90% on what we know about this. 00:37:27.213 --> 00:37:29.380 align:middle line:84% So to get some intuition, let's look at this picture 00:37:29.380 --> 00:37:31.280 align:middle line:84% that I haven't managed to erase yet. 00:37:31.280 --> 00:37:36.420 align:middle line:90% 00:37:36.420 --> 00:37:37.380 align:middle line:90% All right. 00:37:37.380 --> 00:37:40.460 align:middle line:84% So we're trying to understand this thing, the integral over x 00:37:40.460 --> 00:37:42.380 align:middle line:90% of fr squared. 00:37:42.380 --> 00:37:45.747 align:middle line:84% So let's add an x to our picture. 00:37:45.747 --> 00:37:46.830 align:middle line:90% Do I have any more colors? 00:37:46.830 --> 00:37:50.086 align:middle line:90% 00:37:50.086 --> 00:37:55.140 align:middle line:84% So x in this picture is going to be yellow. 00:37:55.140 --> 00:37:57.750 align:middle line:84% And x may look something like this. 00:37:57.750 --> 00:38:08.550 align:middle line:90% 00:38:08.550 --> 00:38:10.750 align:middle line:84% It depends a little bit on Nx of r. 00:38:10.750 --> 00:38:14.370 align:middle line:84% If Nx of r, the biggest that Nx of r could be is r squared. 00:38:14.370 --> 00:38:18.510 align:middle line:84% If it's that big, then x could fill in this whole thing. 00:38:18.510 --> 00:38:20.552 align:middle line:84% But if Nx of r is smaller than r squared, 00:38:20.552 --> 00:38:22.510 align:middle line:84% then it's sort of like the picture that I drew, 00:38:22.510 --> 00:38:26.750 align:middle line:84% and x is only a small fraction of this ball. 00:38:26.750 --> 00:38:30.670 align:middle line:84% And in this picture, it would be lossy to estimate the integral 00:38:30.670 --> 00:38:34.950 align:middle line:84% over just x of fr squared by the global integral of fr 00:38:34.950 --> 00:38:37.130 align:middle line:84% squared, which would include the whole orange ball. 00:38:37.130 --> 00:38:40.430 align:middle line:90% 00:38:40.430 --> 00:38:46.590 align:middle line:84% OK, so we're going to try to do a bit better than that by using 00:38:46.590 --> 00:38:52.070 align:middle line:84% our lemmas about the locally-constant property 00:38:52.070 --> 00:39:01.460 align:middle line:84% of the function fr. All right, I'm going to erase this. 00:39:01.460 --> 00:39:01.960 align:middle line:90% All right. 00:39:01.960 --> 00:39:04.420 align:middle line:84% I'm going to leave this picture here for our intuition. 00:39:04.420 --> 00:39:07.600 align:middle line:90% 00:39:07.600 --> 00:39:08.600 align:middle line:90% All right. 00:39:08.600 --> 00:39:13.120 align:middle line:84% So we have integral over x of fr squared. 00:39:13.120 --> 00:39:17.870 align:middle line:84% I'll just rewrite it as the integral of 1x fr squared. 00:39:17.870 --> 00:39:20.880 align:middle line:90% 00:39:20.880 --> 00:39:24.680 align:middle line:84% OK, now if I want to use the locally-constant property of fr 00:39:24.680 --> 00:39:27.820 align:middle line:84% squared, it means I could use these lemmas. 00:39:27.820 --> 00:39:29.440 align:middle line:84% And in particular, I can use lemma 2, 00:39:29.440 --> 00:39:32.960 align:middle line:84% which is sort of well set up to deal with the square. 00:39:32.960 --> 00:39:39.240 align:middle line:84% So this is bounded by the integral of 1x times fr squared 00:39:39.240 --> 00:39:42.090 align:middle line:90% convolved with cr. 00:39:42.090 --> 00:39:48.080 align:middle line:90% 00:39:48.080 --> 00:39:52.360 align:middle line:84% And if you write out what this is and do Fubini, 00:39:52.360 --> 00:39:55.770 align:middle line:84% this is the same as the integral of fr 00:39:55.770 --> 00:40:00.360 align:middle line:84% squared times the characteristic function of x convolved with cr. 00:40:00.360 --> 00:40:07.730 align:middle line:90% 00:40:07.730 --> 00:40:10.462 align:middle line:90% Do you see that? 00:40:10.462 --> 00:40:11.410 align:middle line:90% Yeah, OK. 00:40:11.410 --> 00:40:12.270 align:middle line:90% Let's do it. 00:40:12.270 --> 00:40:14.450 align:middle line:90% So the algebra is like this. 00:40:14.450 --> 00:40:18.090 align:middle line:90% I have the integral 1x of x. 00:40:18.090 --> 00:40:19.990 align:middle line:84% And then I have this convolution, 00:40:19.990 --> 00:40:27.365 align:middle line:84% which is the integral fr squared of x minus y cr of y dy. 00:40:27.365 --> 00:40:31.944 align:middle line:90% 00:40:31.944 --> 00:40:33.550 align:middle line:90% No, let me do it the other way. 00:40:33.550 --> 00:40:43.610 align:middle line:90% 00:40:43.610 --> 00:40:48.250 align:middle line:84% OK, so now what happens if I Fubini that? 00:40:48.250 --> 00:40:59.900 align:middle line:84% That's the integral fr squared of y integral of 1x of x cr of x 00:40:59.900 --> 00:41:01.526 align:middle line:90% minus y dy. 00:41:01.526 --> 00:41:04.980 align:middle line:90% 00:41:04.980 --> 00:41:07.780 align:middle line:84% OK, it doesn't look quite right yet. 00:41:07.780 --> 00:41:09.912 align:middle line:84% And it actually reminds me that there 00:41:09.912 --> 00:41:12.370 align:middle line:84% was a little something else that's helpful to say about cr. 00:41:12.370 --> 00:41:15.700 align:middle line:90% 00:41:15.700 --> 00:41:20.660 align:middle line:84% So you can also add here that cr is radial. 00:41:20.660 --> 00:41:24.135 align:middle line:90% Implies that it's symmetric. 00:41:24.135 --> 00:41:26.260 align:middle line:84% Yeah, I should probably have mentioned that earlier 00:41:26.260 --> 00:41:27.433 align:middle line:90% in my plan. 00:41:27.433 --> 00:41:29.100 align:middle line:84% So the reason you can do that, remember, 00:41:29.100 --> 00:41:32.260 align:middle line:84% cr was the Fourier transform of some bump function eta. 00:41:32.260 --> 00:41:33.980 align:middle line:84% If your bump function eta is radial, 00:41:33.980 --> 00:41:36.620 align:middle line:84% its Fourier transform will be radial. 00:41:36.620 --> 00:41:38.900 align:middle line:84% So this guy is radial and symmetric. 00:41:38.900 --> 00:41:44.220 align:middle line:84% OK, and then because that's symmetric, I can switch those. 00:41:44.220 --> 00:41:49.820 align:middle line:84% So that's the integral fr squared of y integral 1x of x, 00:41:49.820 --> 00:41:52.830 align:middle line:90% cr of y minus x. 00:41:52.830 --> 00:41:54.610 align:middle line:90% Sorry, this one is dx. 00:41:54.610 --> 00:41:57.930 align:middle line:90% 00:41:57.930 --> 00:41:58.430 align:middle line:90% All right. 00:41:58.430 --> 00:42:05.270 align:middle line:84% So that's the integral of fr squared of y 1x convolved with 00:42:05.270 --> 00:42:07.090 align:middle line:90% cr of y. 00:42:07.090 --> 00:42:29.270 align:middle line:90% 00:42:29.270 --> 00:42:31.590 align:middle line:84% I feel like there also should be a nice image of what's 00:42:31.590 --> 00:42:34.350 align:middle line:90% going on in this integral. 00:42:34.350 --> 00:42:36.590 align:middle line:84% We have this-- imagine for a second 00:42:36.590 --> 00:42:38.650 align:middle line:84% that this was the characteristic function a set. 00:42:38.650 --> 00:42:41.150 align:middle line:84% It's not quite true, but it's not far off either. 00:42:41.150 --> 00:42:44.270 align:middle line:84% So I have one set, and I have another set. 00:42:44.270 --> 00:42:46.610 align:middle line:84% But before I multiply them together, 00:42:46.610 --> 00:42:48.630 align:middle line:84% which would mean intersecting the sets, 00:42:48.630 --> 00:42:52.560 align:middle line:84% I fatten out this set here by a distance r. 00:42:52.560 --> 00:42:56.920 align:middle line:84% So I take this, I call this set y, take y, I fatten it out by r, 00:42:56.920 --> 00:42:59.560 align:middle line:90% and I intersect it with x. 00:42:59.560 --> 00:43:01.120 align:middle line:84% And that's kind of the same thing 00:43:01.120 --> 00:43:02.920 align:middle line:84% as taking x and fattening it out by r, 00:43:02.920 --> 00:43:05.622 align:middle line:84% and intersecting it with y, because I could describe 00:43:05.622 --> 00:43:06.580 align:middle line:90% both of them like this. 00:43:06.580 --> 00:43:09.360 align:middle line:84% I'm looking for pairs of points, 1 in x and 1 and y. 00:43:09.360 --> 00:43:11.440 align:middle line:84% The distance between them is at most r. 00:43:11.440 --> 00:43:14.860 align:middle line:84% If I say it that way, it's symmetric. 00:43:14.860 --> 00:43:16.420 align:middle line:90% And that's what's going on here. 00:43:16.420 --> 00:43:19.840 align:middle line:90% 00:43:19.840 --> 00:43:23.520 align:middle line:84% Yeah, I guess that's a fancy way of saying, 00:43:23.520 --> 00:43:27.160 align:middle line:84% I just took this integral and I wrote it like this. 00:43:27.160 --> 00:43:31.320 align:middle line:84% And now, remember that the function, cr, is symmetric. 00:43:31.320 --> 00:43:34.580 align:middle line:84% Now the whole thing is symmetric, symmetric 00:43:34.580 --> 00:43:36.340 align:middle line:84% in exchanging these two functions. 00:43:36.340 --> 00:43:39.760 align:middle line:90% 00:43:39.760 --> 00:43:42.360 align:middle line:90% OK, cool. 00:43:42.360 --> 00:43:44.180 align:middle line:84% OK, so now, we've written it this way. 00:43:44.180 --> 00:43:46.840 align:middle line:84% And the intuitive meaning of this 00:43:46.840 --> 00:43:49.200 align:middle line:84% is I've averaged the characteristic function 00:43:49.200 --> 00:43:51.510 align:middle line:90% of x over balls of radius r. 00:43:51.510 --> 00:43:54.290 align:middle line:90% 00:43:54.290 --> 00:44:02.530 align:middle line:84% And so this is bounded by Nx of r over r squared. 00:44:02.530 --> 00:44:07.897 align:middle line:84% So this is the integral of that guy over a ball of radius r. 00:44:07.897 --> 00:44:09.230 align:middle line:90% And then I'm taking the average. 00:44:09.230 --> 00:44:13.750 align:middle line:84% So I divide by the area of the ball of radius r. 00:44:13.750 --> 00:44:14.250 align:middle line:90% OK. 00:44:14.250 --> 00:44:17.610 align:middle line:90% 00:44:17.610 --> 00:44:20.350 align:middle line:84% So to check this a little bit more carefully, 00:44:20.350 --> 00:44:23.930 align:middle line:84% we have to use the bounds that we have for this cr thing, which 00:44:23.930 --> 00:44:24.750 align:middle line:90% are over here. 00:44:24.750 --> 00:44:27.610 align:middle line:90% 00:44:27.610 --> 00:44:32.210 align:middle line:84% So it's bounded by 1 over r squared on the ball of radius r. 00:44:32.210 --> 00:44:33.850 align:middle line:90% That would give the average. 00:44:33.850 --> 00:44:36.290 align:middle line:84% Then it also has this rapidly-decaying tail, 00:44:36.290 --> 00:44:38.930 align:middle line:84% which to be careful, we should take account of. 00:44:38.930 --> 00:44:43.990 align:middle line:84% But what it contributes is smaller than the first term, 00:44:43.990 --> 00:44:44.670 align:middle line:90% the main term. 00:44:44.670 --> 00:44:49.060 align:middle line:90% 00:44:49.060 --> 00:44:52.060 align:middle line:90% OK, so that's how we bound this. 00:44:52.060 --> 00:44:53.060 align:middle line:90% This is now a number. 00:44:53.060 --> 00:44:54.920 align:middle line:84% So we can take it out of the integral. 00:44:54.920 --> 00:44:57.440 align:middle line:84% And we just have left the L2 norm of fr squared. 00:44:57.440 --> 00:44:59.860 align:middle line:84% So we plug in our bound for that. 00:44:59.860 --> 00:45:04.440 align:middle line:84% So we have, it's bounded by Nx of r over r squared, 00:45:04.440 --> 00:45:08.540 align:middle line:84% you can think of this thing as the density of x at scale r, 00:45:08.540 --> 00:45:11.800 align:middle line:84% times the integral of fr squared OK, 00:45:11.800 --> 00:45:16.140 align:middle line:84% and then I just plug in the bound for the integral of fr 00:45:16.140 --> 00:45:16.860 align:middle line:90% squared. 00:45:16.860 --> 00:45:19.180 align:middle line:90% And we'll write that out. 00:45:19.180 --> 00:45:22.860 align:middle line:90% So OK, we have number of tubes. 00:45:22.860 --> 00:45:25.980 align:middle line:90% We have big r over little r. 00:45:25.980 --> 00:45:29.060 align:middle line:90% We have Nt of r. 00:45:29.060 --> 00:45:30.780 align:middle line:90% That's this thing. 00:45:30.780 --> 00:45:34.300 align:middle line:90% Then we have Nx of r. 00:45:34.300 --> 00:45:35.630 align:middle line:90% And then we have r squared. 00:45:35.630 --> 00:45:40.740 align:middle line:90% 00:45:40.740 --> 00:45:44.380 align:middle line:90% OK, we are almost done. 00:45:44.380 --> 00:45:51.910 align:middle line:84% But what we have here is number of tubes in a little-r-by-big-R 00:45:51.910 --> 00:45:52.510 align:middle line:90% guy. 00:45:52.510 --> 00:45:54.670 align:middle line:84% And the information we want to be using 00:45:54.670 --> 00:45:56.990 align:middle line:90% is the number of directions. 00:45:56.990 --> 00:46:01.590 align:middle line:84% So the last step is that the number of tubes 00:46:01.590 --> 00:46:04.830 align:middle line:84% in the little-r-by-big-R tube is bounded by the number 00:46:04.830 --> 00:46:09.710 align:middle line:84% of directions in a little-r-over-big-R angle times 00:46:09.710 --> 00:46:10.630 align:middle line:90% little r. 00:46:10.630 --> 00:46:12.990 align:middle line:90% So here's a picture. 00:46:12.990 --> 00:46:16.710 align:middle line:84% Here's our little-r-by-big-R tube. 00:46:16.710 --> 00:46:22.510 align:middle line:84% Inside of it, I can fit in some tubes in different directions. 00:46:22.510 --> 00:46:26.750 align:middle line:84% So if I pick one direction, I can pick little r of them. 00:46:26.750 --> 00:46:28.270 align:middle line:90% That's this little r. 00:46:28.270 --> 00:46:30.990 align:middle line:84% But they don't have to go that way. 00:46:30.990 --> 00:46:35.250 align:middle line:84% I could also have them go maybe the most extreme directions. 00:46:35.250 --> 00:46:39.190 align:middle line:90% I could have them go that way. 00:46:39.190 --> 00:46:44.280 align:middle line:90% And this angle here is little r. 00:46:44.280 --> 00:46:47.710 align:middle line:84% The angle there is little r over big. 00:46:47.710 --> 00:46:50.280 align:middle line:90% 00:46:50.280 --> 00:46:55.360 align:middle line:84% So all of the different tubes that appear here, 00:46:55.360 --> 00:46:57.800 align:middle line:84% they have to have an angle within a range 00:46:57.800 --> 00:47:00.360 align:middle line:84% of little r over big R. So the number of directions 00:47:00.360 --> 00:47:02.080 align:middle line:90% is like that. 00:47:02.080 --> 00:47:03.960 align:middle line:84% And then for each direction, then the number 00:47:03.960 --> 00:47:07.632 align:middle line:84% of possible tubes is at most, little r. 00:47:07.632 --> 00:47:11.000 align:middle line:84% OK, so if you substitute this for that, 00:47:11.000 --> 00:47:14.220 align:middle line:84% then this expression becomes the right-hand side of theorem two. 00:47:14.220 --> 00:47:31.140 align:middle line:90% 00:47:31.140 --> 00:47:31.640 align:middle line:90% Yeah. 00:47:31.640 --> 00:47:33.640 align:middle line:84% AUDIENCE: I remember kind of use the fact 00:47:33.640 --> 00:47:35.350 align:middle line:84% that the tubes that are at an angle, 00:47:35.350 --> 00:47:38.840 align:middle line:84% that you can only fit less of them in. 00:47:38.840 --> 00:47:43.690 align:middle line:84% Does that change the boundary of anything different? 00:47:43.690 --> 00:47:44.830 align:middle line:90% LAWRENCE GUTH: Yeah, OK. 00:47:44.830 --> 00:47:50.510 align:middle line:84% So the question was if we knew that in a given direction, 00:47:50.510 --> 00:47:54.770 align:middle line:84% these tubes had to have some spacing, 00:47:54.770 --> 00:47:56.970 align:middle line:84% so we could also measure the clustering of the tubes 00:47:56.970 --> 00:47:58.230 align:middle line:90% in a given direction. 00:47:58.230 --> 00:48:01.050 align:middle line:84% So we add some information about that, then maybe this 00:48:01.050 --> 00:48:02.870 align:middle line:84% would have to be smaller than little r. 00:48:02.870 --> 00:48:07.090 align:middle line:90% 00:48:07.090 --> 00:48:08.750 align:middle line:84% So our setup involves three things. 00:48:08.750 --> 00:48:11.850 align:middle line:84% But we only thought about the clustering of two of them. 00:48:11.850 --> 00:48:15.250 align:middle line:84% Our setup involves a set of unit balls, x, 00:48:15.250 --> 00:48:17.930 align:middle line:84% and we looked at how it's clustered, Nx of r. 00:48:17.930 --> 00:48:20.330 align:middle line:84% Set of directions, D. And we look 00:48:20.330 --> 00:48:22.730 align:middle line:84% at how it's clustered, Nd of rho. 00:48:22.730 --> 00:48:26.428 align:middle line:84% And also, a set of projections, which each have size 00:48:26.428 --> 00:48:27.970 align:middle line:84% at most S. But we didn't say anything 00:48:27.970 --> 00:48:30.290 align:middle line:90% about how they were clustered. 00:48:30.290 --> 00:48:32.210 align:middle line:84% So you could add to this setup something 00:48:32.210 --> 00:48:35.250 align:middle line:84% about how each projection is clustered. 00:48:35.250 --> 00:48:41.300 align:middle line:84% And you could then use that to improve this thing here. 00:48:41.300 --> 00:48:41.800 align:middle line:90% Yeah. 00:48:41.800 --> 00:48:43.550 align:middle line:84% So that could be a little homework project 00:48:43.550 --> 00:48:45.120 align:middle line:84% if anyone is interested in doing it. 00:48:45.120 --> 00:48:50.340 align:middle line:90% 00:48:50.340 --> 00:48:52.777 align:middle line:90% Yeah, OK. 00:48:52.777 --> 00:48:54.110 align:middle line:90% Any other questions or comments? 00:48:54.110 --> 00:49:00.580 align:middle line:90% 00:49:00.580 --> 00:49:05.750 align:middle line:90% OK, OK. 00:49:05.750 --> 00:49:07.660 align:middle line:84% So this theorem sort of naturally 00:49:07.660 --> 00:49:11.860 align:middle line:84% falls out from using the Fourier method. 00:49:11.860 --> 00:49:14.060 align:middle line:90% But it's a little messy looking. 00:49:14.060 --> 00:49:16.660 align:middle line:84% And the thing that's particularly messy looking 00:49:16.660 --> 00:49:20.380 align:middle line:84% is this maximum over little r, and x, and d, da, da, da. 00:49:20.380 --> 00:49:22.500 align:middle line:84% And there's a fairly general situation 00:49:22.500 --> 00:49:25.460 align:middle line:84% that we mentioned last time, where that messy bit 00:49:25.460 --> 00:49:27.820 align:middle line:90% gets a lot cleaner. 00:49:27.820 --> 00:49:30.120 align:middle line:84% I don't know that that's so philosophically important, 00:49:30.120 --> 00:49:31.680 align:middle line:84% but I think it's worth mentioning. 00:49:31.680 --> 00:49:33.860 align:middle line:84% And this is the way the results usually 00:49:33.860 --> 00:49:35.471 align:middle line:90% appear in the literature. 00:49:35.471 --> 00:49:38.060 align:middle line:90% 00:49:38.060 --> 00:49:38.800 align:middle line:90% All right. 00:49:38.800 --> 00:49:44.270 align:middle line:90% 00:49:44.270 --> 00:49:48.350 align:middle line:84% And this will allow us to finally prove the result 00:49:48.350 --> 00:49:49.690 align:middle line:90% that we stated last time. 00:49:49.690 --> 00:50:01.710 align:middle line:90% 00:50:01.710 --> 00:50:04.390 align:middle line:90% All right. 00:50:04.390 --> 00:50:11.650 align:middle line:84% So definition, we say x is Hausdorff, or Hausdorff spacing. 00:50:11.650 --> 00:50:14.415 align:middle line:90% 00:50:14.415 --> 00:50:19.390 align:middle line:84% f and x of r to the beta is bounded 00:50:19.390 --> 00:50:27.550 align:middle line:84% by around x to the beta for all beta between 0 and 1. 00:50:27.550 --> 00:50:31.150 align:middle line:84% So when beta is 1, this is trivial. 00:50:31.150 --> 00:50:32.170 align:middle line:90% This is always true. 00:50:32.170 --> 00:50:34.330 align:middle line:84% But this is saying that at smaller scales beta, 00:50:34.330 --> 00:50:36.870 align:middle line:84% this x is not too clumped compared 00:50:36.870 --> 00:50:39.820 align:middle line:90% to how it is at the top scale. 00:50:39.820 --> 00:50:41.360 align:middle line:90% Yeah. 00:50:41.360 --> 00:50:43.318 align:middle line:84% AUDIENCE: How is the [INAUDIBLE]? 00:50:43.318 --> 00:50:47.230 align:middle line:90% 00:50:47.230 --> 00:50:48.760 align:middle line:90% LAWRENCE GUTH: Ah, sorry. 00:50:48.760 --> 00:50:51.440 align:middle line:90% Thanks. 00:50:51.440 --> 00:50:56.200 align:middle line:84% Yeah, there was one other thing I forgot to do. 00:50:56.200 --> 00:50:57.500 align:middle line:90% OK, right. 00:50:57.500 --> 00:50:58.000 align:middle line:90% Sorry. 00:50:58.000 --> 00:50:58.500 align:middle line:90% Thanks. 00:50:58.500 --> 00:51:00.800 align:middle line:84% So OK, so the question is, how did we 00:51:00.800 --> 00:51:03.080 align:middle line:84% actually do all this algebra and get the equation 00:51:03.080 --> 00:51:05.597 align:middle line:84% that the inequality is actually stated in theorem two? 00:51:05.597 --> 00:51:07.680 align:middle line:84% And thank you for the question, because I left out 00:51:07.680 --> 00:51:10.760 align:middle line:90% some of the steps. 00:51:10.760 --> 00:51:15.680 align:middle line:84% OK, so now that we have done the main ideas, 00:51:15.680 --> 00:51:18.380 align:middle line:84% we're just talking about the algebra, I will erase this. 00:51:18.380 --> 00:51:22.600 align:middle line:90% 00:51:22.600 --> 00:51:25.880 align:middle line:90% And so what do we have so far? 00:51:25.880 --> 00:51:28.160 align:middle line:84% If you put together the inequalities we have so far, 00:51:28.160 --> 00:51:31.540 align:middle line:84% we have x times the number of directions squared. 00:51:31.540 --> 00:51:33.140 align:middle line:84% That's where we started over here. 00:51:33.140 --> 00:51:37.250 align:middle line:84% Then we did some stuff, and we ended up here. 00:51:37.250 --> 00:51:40.610 align:middle line:84% So that's bounded by number of tubes 00:51:40.610 --> 00:51:52.890 align:middle line:84% times big R over little r times Nd little r over big R times r. 00:51:52.890 --> 00:51:55.570 align:middle line:90% That was this thing for there. 00:51:55.570 --> 00:51:59.010 align:middle line:84% And x of r over little r squared. 00:51:59.010 --> 00:52:02.650 align:middle line:90% And we can cross those guys. 00:52:02.650 --> 00:52:04.810 align:middle line:84% Now we still have this number of tubes. 00:52:04.810 --> 00:52:06.090 align:middle line:90% What's that? 00:52:06.090 --> 00:52:11.050 align:middle line:84% Well, we have S tubes in each direction, we have D directions. 00:52:11.050 --> 00:52:15.970 align:middle line:84% So this is S times D. OK, so now, 00:52:15.970 --> 00:52:17.830 align:middle line:84% I've gotten rid of all the extra letters. 00:52:17.830 --> 00:52:19.450 align:middle line:84% And we only have left the letters 00:52:19.450 --> 00:52:21.930 align:middle line:84% that appear in the inequality we want. 00:52:21.930 --> 00:52:24.660 align:middle line:84% And then the last thing you do is you get D by itself. 00:52:24.660 --> 00:52:28.930 align:middle line:90% 00:52:28.930 --> 00:52:31.130 align:middle line:84% So I take one factor of D, and bring it over there, 00:52:31.130 --> 00:52:32.370 align:middle line:90% and cancel the two. 00:52:32.370 --> 00:52:34.620 align:middle line:84% Then I take the x and bring it over here. 00:52:34.620 --> 00:52:37.320 align:middle line:84% And then what's left is what's there. 00:52:37.320 --> 00:52:40.540 align:middle line:90% 00:52:40.540 --> 00:52:41.260 align:middle line:90% Yeah. 00:52:41.260 --> 00:52:43.620 align:middle line:84% AUDIENCE: And that's inequality be made sharp 00:52:43.620 --> 00:52:46.220 align:middle line:90% with any [INAUDIBLE]? 00:52:46.220 --> 00:52:55.707 align:middle line:84% Like in any choice of S, x, and D [INAUDIBLE]? 00:52:55.707 --> 00:52:56.540 align:middle line:90% LAWRENCE GUTH: Yeah. 00:52:56.540 --> 00:53:01.940 align:middle line:90% 00:53:01.940 --> 00:53:02.680 align:middle line:90% Yes. 00:53:02.680 --> 00:53:06.860 align:middle line:84% OK, so the question is, is this inequality ever sharp? 00:53:06.860 --> 00:53:12.080 align:middle line:84% And it is sometimes sharp, although not super often. 00:53:12.080 --> 00:53:16.060 align:middle line:84% And the cases that I have worked out so far when it's sharp 00:53:16.060 --> 00:53:19.060 align:middle line:84% are within this Hausdorff setting. 00:53:19.060 --> 00:53:20.120 align:middle line:90% So let's maybe do that. 00:53:20.120 --> 00:53:21.480 align:middle line:90% It'll all get a lot simpler. 00:53:21.480 --> 00:53:23.520 align:middle line:84% And then I can say what the cases I know are. 00:53:23.520 --> 00:53:26.900 align:middle line:90% 00:53:26.900 --> 00:53:32.340 align:middle line:84% OK, so the Hausdorff spacing is a bound on Nx of little 00:53:32.340 --> 00:53:35.710 align:middle line:84% r in terms of the size of x that looks like this. 00:53:35.710 --> 00:53:38.710 align:middle line:84% And I coined the term, Hausdorff spacing, 00:53:38.710 --> 00:53:41.470 align:middle line:84% because it comes up naturally when you talk about Hausdorff 00:53:41.470 --> 00:53:44.270 align:middle line:90% measures and things like that. 00:53:44.270 --> 00:53:46.790 align:middle line:84% OK, and it's useful in our setting 00:53:46.790 --> 00:53:56.750 align:middle line:84% because if x and D have Hausdorff spacing, then 00:53:56.750 --> 00:54:03.210 align:middle line:84% when you take the maximum of, actually many expressions, 00:54:03.210 --> 00:54:11.870 align:middle line:84% but the one that comes up for us here is this, OK, what happens? 00:54:11.870 --> 00:54:14.890 align:middle line:84% So I assume that both x and D have Hausdorff spacing. 00:54:14.890 --> 00:54:17.070 align:middle line:84% It's important to have both of them. 00:54:17.070 --> 00:54:18.910 align:middle line:84% Then each one of these things get 00:54:18.910 --> 00:54:24.470 align:middle line:84% replaced by some power of x or D. 00:54:24.470 --> 00:54:31.993 align:middle line:84% And once you have a power law, so the whole expression will be, 00:54:31.993 --> 00:54:34.160 align:middle line:84% or another way to look at it is the whole expression 00:54:34.160 --> 00:54:36.010 align:middle line:90% will be some power of little r. 00:54:36.010 --> 00:54:37.860 align:middle line:84% A power could be positive or negative, 00:54:37.860 --> 00:54:43.080 align:middle line:84% but the maximum will occur either at 1 or at big R. 00:54:43.080 --> 00:54:47.910 align:middle line:84% So this guy basically is equal to what happens at 1. 00:54:47.910 --> 00:54:49.520 align:middle line:90% Nx of 1 is always 1. 00:54:49.520 --> 00:54:52.160 align:middle line:90% ND of 1 over r is always 1. 00:54:52.160 --> 00:54:53.880 align:middle line:90% So we just get 1. 00:54:53.880 --> 00:54:58.880 align:middle line:84% And when little r is big R, and x of big r is all the points, 00:54:58.880 --> 00:55:02.620 align:middle line:84% MD of 1 is all the directions, And we have that. 00:55:02.620 --> 00:55:06.000 align:middle line:90% 00:55:06.000 --> 00:55:07.720 align:middle line:90% So it's less nasty. 00:55:07.720 --> 00:55:11.920 align:middle line:84% And so we get the following corollary. 00:55:11.920 --> 00:55:16.040 align:middle line:84% So corollary says if we have our setup, 00:55:16.040 --> 00:55:23.880 align:middle line:84% and x and D have Hausdorff spacing, 00:55:23.880 --> 00:55:34.410 align:middle line:84% then the number of directions is bounded by SR 00:55:34.410 --> 00:55:41.050 align:middle line:84% over x times 1 plus SR over x times this thing. 00:55:41.050 --> 00:55:45.110 align:middle line:90% So SD over r. 00:55:45.110 --> 00:55:49.170 align:middle line:90% 00:55:49.170 --> 00:55:51.930 align:middle line:84% OK, now this one has an interpretation. 00:55:51.930 --> 00:55:55.570 align:middle line:84% If this term dominates, you can cancel the D's, and it tells you 00:55:55.570 --> 00:55:58.330 align:middle line:90% that S is almost r. 00:55:58.330 --> 00:56:02.650 align:middle line:84% So we get either S is almost as big 00:56:02.650 --> 00:56:04.790 align:middle line:84% as R. R is the biggest it could possibly be. 00:56:04.790 --> 00:56:08.970 align:middle line:84% So S is almost as big as it could possibly be. 00:56:08.970 --> 00:56:15.050 align:middle line:90% Or D is bounded by SR over x. 00:56:15.050 --> 00:56:17.630 align:middle line:84% So that's what we stated last time. 00:56:17.630 --> 00:56:21.930 align:middle line:84% The formula is not as nasty as the general formula. 00:56:21.930 --> 00:56:22.850 align:middle line:90% Yeah. 00:56:22.850 --> 00:56:26.610 align:middle line:84% AUDIENCE: There's also a factor of log [INAUDIBLE] theorem two, 00:56:26.610 --> 00:56:28.108 align:middle line:90% because you're choosing R. 00:56:28.108 --> 00:56:28.900 align:middle line:90% LAWRENCE GUTH: Yep. 00:56:28.900 --> 00:56:29.400 align:middle line:90% Thanks. 00:56:29.400 --> 00:56:32.660 align:middle line:90% 00:56:32.660 --> 00:56:35.820 align:middle line:84% Yeah, so the point from the audience 00:56:35.820 --> 00:56:39.220 align:middle line:84% was that there was an extra factor of log R 00:56:39.220 --> 00:56:40.800 align:middle line:90% that I forgot to write down. 00:56:40.800 --> 00:56:43.100 align:middle line:84% So I'll put this to represent that. 00:56:43.100 --> 00:56:45.740 align:middle line:90% Yeah, thanks. 00:56:45.740 --> 00:56:48.300 align:middle line:90% OK. 00:56:48.300 --> 00:56:52.260 align:middle line:84% OK, so this relatively clean estimate, 00:56:52.260 --> 00:56:56.100 align:middle line:84% it exactly matches our theorem two in finite fields. 00:56:56.100 --> 00:57:00.060 align:middle line:84% I won't write it down, but the numerology is the same. 00:57:00.060 --> 00:57:01.960 align:middle line:84% And it's a little nicer to look at. 00:57:01.960 --> 00:57:04.180 align:middle line:90% It still takes some digesting. 00:57:04.180 --> 00:57:08.600 align:middle line:90% And this is sharp in some cases. 00:57:08.600 --> 00:57:12.620 align:middle line:84% It's sharp in the case that S is pretty close to R. 00:57:12.620 --> 00:57:15.340 align:middle line:84% So because I need this, the smallest I could make 00:57:15.340 --> 00:57:18.300 align:middle line:84% S would be something like R over log R. 00:57:18.300 --> 00:57:20.940 align:middle line:84% And if you take S around that, R over log R, 00:57:20.940 --> 00:57:23.540 align:middle line:84% or R to the 1 minus epsilon, then this 00:57:23.540 --> 00:57:26.740 align:middle line:84% is sharp in the example of a grid of points, 00:57:26.740 --> 00:57:28.010 align:middle line:90% grid of unit balls. 00:57:28.010 --> 00:57:31.190 align:middle line:90% 00:57:31.190 --> 00:57:33.870 align:middle line:84% Yeah, so that's the main interesting example when-- 00:57:33.870 --> 00:57:37.230 align:middle line:84% that's the main example that I know when it's sharp. 00:57:37.230 --> 00:57:40.050 align:middle line:84% And there are plenty of examples where it's not sharp. 00:57:40.050 --> 00:57:43.590 align:middle line:84% But that grid example is important. 00:57:43.590 --> 00:57:44.670 align:middle line:90% Yeah. 00:57:44.670 --> 00:57:46.850 align:middle line:84% OK, I guess it's also a natural question, 00:57:46.850 --> 00:57:50.230 align:middle line:84% if we go to the regime that's not Hausdorff, 00:57:50.230 --> 00:57:54.870 align:middle line:84% this theorem added some junk here 00:57:54.870 --> 00:57:58.630 align:middle line:84% to account for things being more clustered. 00:57:58.630 --> 00:58:00.910 align:middle line:84% And it's a natural question, is there 00:58:00.910 --> 00:58:05.428 align:middle line:84% any scenario where this junk is exactly sharp? 00:58:05.428 --> 00:58:07.970 align:middle line:84% And I wondered about that when I was working on this lecture. 00:58:07.970 --> 00:58:09.610 align:middle line:90% But I don't know the answer yet. 00:58:09.610 --> 00:58:11.035 align:middle line:84% So if anybody wants to try, that's 00:58:11.035 --> 00:58:12.160 align:middle line:90% also a nice little project. 00:58:12.160 --> 00:58:22.670 align:middle line:90% 00:58:22.670 --> 00:58:25.360 align:middle line:90% Cool. 00:58:25.360 --> 00:58:28.020 align:middle line:84% OK, so that was the meat of the class. 00:58:28.020 --> 00:58:35.400 align:middle line:84% And that was my discussion that I planned about the Fourier 00:58:35.400 --> 00:58:37.160 align:middle line:90% method in the real case. 00:58:37.160 --> 00:58:40.520 align:middle line:84% So it's another good moment to see if you 00:58:40.520 --> 00:58:42.903 align:middle line:90% have questions to digest. 00:58:42.903 --> 00:58:44.820 align:middle line:84% Then we'll start something a little different. 00:58:44.820 --> 00:58:52.560 align:middle line:90% 00:58:52.560 --> 00:58:53.960 align:middle line:90% Yeah. 00:58:53.960 --> 00:58:58.200 align:middle line:84% AUDIENCE: For the maximum term, could you explain why it has 00:58:58.200 --> 00:59:01.360 align:middle line:90% to be a maximum and it's-- 00:59:01.360 --> 00:59:02.920 align:middle line:84% the part of the proof that requires 00:59:02.920 --> 00:59:06.300 align:middle line:84% taking the maximum over, or instead of, any given r? 00:59:06.300 --> 00:59:08.910 align:middle line:90% 00:59:08.910 --> 00:59:10.660 align:middle line:84% LAWRENCE GUTH: So I think the question is, 00:59:10.660 --> 00:59:12.920 align:middle line:84% why do we have a maximum over r instead 00:59:12.920 --> 00:59:14.800 align:middle line:90% of maybe like a sum over r? 00:59:14.800 --> 00:59:15.497 align:middle line:90% AUDIENCE: Yeah. 00:59:15.497 --> 00:59:16.330 align:middle line:90% LAWRENCE GUTH: Yeah. 00:59:16.330 --> 00:59:18.400 align:middle line:84% AUDIENCE: Just where does the maximum come from? 00:59:18.400 --> 00:59:20.960 align:middle line:90% LAWRENCE GUTH: Yeah. 00:59:20.960 --> 00:59:23.520 align:middle line:84% So you could put sum over little r. 00:59:23.520 --> 00:59:25.370 align:middle line:84% And the difference isn't very important, 00:59:25.370 --> 00:59:27.770 align:middle line:84% because the number of terms is only log of r. 00:59:27.770 --> 00:59:30.530 align:middle line:84% And the log of r has got absorbed here. 00:59:30.530 --> 00:59:32.530 align:middle line:84% But where did this come from at all? 00:59:32.530 --> 00:59:35.410 align:middle line:84% Well, we had this function that's 00:59:35.410 --> 00:59:38.130 align:middle line:84% the sum of all these tubes, which describes our projection 00:59:38.130 --> 00:59:39.530 align:middle line:90% situation. 00:59:39.530 --> 00:59:41.690 align:middle line:90% And then we are-- 00:59:41.690 --> 00:59:44.663 align:middle line:84% the key thing is that we applied the main lemma to this function. 00:59:44.663 --> 00:59:46.330 align:middle line:84% And the main lemma said you can break up 00:59:46.330 --> 00:59:47.730 align:middle line:90% this function into a few-- 00:59:47.730 --> 00:59:49.170 align:middle line:90% AUDIENCE: [INAUDIBLE] 00:59:49.170 --> 00:59:50.410 align:middle line:90% LAWRENCE GUTH: Yeah. 00:59:50.410 --> 00:59:52.922 align:middle line:84% And so one of those pieces is sort of the important one. 00:59:52.922 --> 00:59:54.880 align:middle line:84% And that's the one that appears in the maximum. 00:59:54.880 --> 00:59:57.750 align:middle line:90% 00:59:57.750 --> 00:59:58.250 align:middle line:90% OK. 00:59:58.250 --> 01:00:01.250 align:middle line:90% 01:00:01.250 --> 01:00:02.250 align:middle line:90% Yeah. 01:00:02.250 --> 01:00:03.730 align:middle line:90% AUDIENCE: Yeah, I'm wondering. 01:00:03.730 --> 01:00:09.090 align:middle line:84% So in the proof, we had to keep track of all the r's. 01:00:09.090 --> 01:00:11.570 align:middle line:84% In the Hausdorff spacing case, it 01:00:11.570 --> 01:00:15.170 align:middle line:84% seems like just there's only two cases. 01:00:15.170 --> 01:00:17.950 align:middle line:84% So I'm wondering if you do have cluster spacing, 01:00:17.950 --> 01:00:19.730 align:middle line:90% is it really necessary? 01:00:19.730 --> 01:00:25.300 align:middle line:84% Is it still important to do the static division? 01:00:25.300 --> 01:00:27.300 align:middle line:84% LAWRENCE GUTH: OK, so the question is, 01:00:27.300 --> 01:00:30.260 align:middle line:84% in the Hausdorff spacing case, it turned out at the end 01:00:30.260 --> 01:00:33.700 align:middle line:84% of the day, that only two values of little r mattered 01:00:33.700 --> 01:00:38.380 align:middle line:84% in the final inequality, namely 1 and capital R. 01:00:38.380 --> 01:00:41.540 align:middle line:84% And the question was, would it be possible to write the proof 01:00:41.540 --> 01:00:45.420 align:middle line:84% so that only those two appeared and we didn't have all this 01:00:45.420 --> 01:00:48.340 align:middle line:84% rather complicated business of keeping track of all 01:00:48.340 --> 01:00:50.900 align:middle line:90% the intermediate little r's? 01:00:50.900 --> 01:00:52.440 align:middle line:90% I don't know any way to do that. 01:00:52.440 --> 01:00:55.260 align:middle line:84% I think the only way I understand 01:00:55.260 --> 01:00:57.660 align:middle line:84% it is to keep track of all the little R's, 01:00:57.660 --> 01:01:01.280 align:middle line:84% and then notice that under these hypotheses, 01:01:01.280 --> 01:01:05.070 align:middle line:84% the intermediate R's are bounded by the extreme ones. 01:01:05.070 --> 01:01:14.340 align:middle line:90% 01:01:14.340 --> 01:01:16.340 align:middle line:90% OK. 01:01:16.340 --> 01:01:19.660 align:middle line:84% OK, so I thought I would say a little bit about the history 01:01:19.660 --> 01:01:20.480 align:middle line:90% of these ideas. 01:01:20.480 --> 01:01:23.510 align:middle line:84% I haven't been putting names next to the theorems. 01:01:23.510 --> 01:01:26.230 align:middle line:84% And the reason is that these theorems 01:01:26.230 --> 01:01:29.990 align:middle line:84% are small variations of theorems from the literature. 01:01:29.990 --> 01:01:32.230 align:middle line:84% And there are many kind of related theorems 01:01:32.230 --> 01:01:33.365 align:middle line:90% from the literature. 01:01:33.365 --> 01:01:35.490 align:middle line:84% But let me tell you a little bit about the history. 01:01:35.490 --> 01:01:36.250 align:middle line:90% I think it's interesting. 01:01:36.250 --> 01:01:38.130 align:middle line:84% I think these are kind of fundamental ideas. 01:01:38.130 --> 01:01:41.093 align:middle line:84% And they were discovered independently, or somewhat 01:01:41.093 --> 01:01:42.510 align:middle line:84% independently, by different people 01:01:42.510 --> 01:01:43.760 align:middle line:90% working on different problems. 01:01:43.760 --> 01:01:56.310 align:middle line:90% 01:01:56.310 --> 01:02:00.510 align:middle line:84% So the thing that we have been calling the Fourier method, 01:02:00.510 --> 01:02:06.990 align:middle line:84% so let's call this history, the thing that we are calling 01:02:06.990 --> 01:02:10.890 align:middle line:84% the Fourier method appears in a bunch places, 01:02:10.890 --> 01:02:12.750 align:middle line:90% including the following. 01:02:12.750 --> 01:02:15.750 align:middle line:84% The earliest that I am aware of is Linnik's work 01:02:15.750 --> 01:02:21.480 align:middle line:84% in sieve theory, which I will introduce sieve theory 01:02:21.480 --> 01:02:25.440 align:middle line:84% and tell you what this work was today. 01:02:25.440 --> 01:02:29.940 align:middle line:90% So that was in the 1940s. 01:02:29.940 --> 01:02:35.240 align:middle line:84% The second place I'm aware of is Roth's work on the Heilbronn 01:02:35.240 --> 01:02:39.840 align:middle line:84% triangle problem, which at some point in the class probably, 01:02:39.840 --> 01:02:43.810 align:middle line:84% we'll talk about a little bit, in the '70s. 01:02:43.810 --> 01:02:47.300 align:middle line:90% 01:02:47.300 --> 01:02:48.297 align:middle line:90% He knew sieve theory. 01:02:48.297 --> 01:02:50.380 align:middle line:84% He knew Linnik's work well, and he'd worked on it. 01:02:50.380 --> 01:02:52.740 align:middle line:84% But in this problem, he did something similar 01:02:52.740 --> 01:02:53.860 align:middle line:90% that's in Euclidean space. 01:02:53.860 --> 01:02:55.790 align:middle line:84% And the setup is kind of different. 01:02:55.790 --> 01:02:57.760 align:middle line:90% And so I put it on the list. 01:02:57.760 --> 01:03:01.120 align:middle line:84% Then it was something like this was done by Falconer 01:03:01.120 --> 01:03:02.570 align:middle line:90% in geometric measure theory. 01:03:02.570 --> 01:03:08.338 align:middle line:90% 01:03:08.338 --> 01:03:08.840 align:middle line:90% All right. 01:03:08.840 --> 01:03:12.240 align:middle line:84% So Falconer essentially proved the corollary 01:03:12.240 --> 01:03:13.880 align:middle line:90% that I just erased. 01:03:13.880 --> 01:03:17.320 align:middle line:90% And that was around 1980. 01:03:17.320 --> 01:03:21.790 align:middle line:84% And then it was done by Vinh in the setting of finite fields. 01:03:21.790 --> 01:03:22.780 align:middle line:90% So lines in Fq2. 01:03:22.780 --> 01:03:28.730 align:middle line:90% 01:03:28.730 --> 01:03:30.490 align:middle line:84% And I'm not positive about this, but I 01:03:30.490 --> 01:03:33.130 align:middle line:84% think that many of these people were not fully aware of others 01:03:33.130 --> 01:03:36.330 align:middle line:90% of these people. 01:03:36.330 --> 01:03:40.390 align:middle line:84% OK, we also mentioned the double counting method. 01:03:40.390 --> 01:03:48.498 align:middle line:90% 01:03:48.498 --> 01:03:52.030 align:middle line:84% And the double counting method that we did, 01:03:52.030 --> 01:03:57.250 align:middle line:84% so it was done by Kaufman in the context of geometric measure 01:03:57.250 --> 01:04:06.050 align:middle line:90% theory in the 1960s. 01:04:06.050 --> 01:04:13.290 align:middle line:84% And it was done by Gallagher in the context of sieve theory I 01:04:13.290 --> 01:04:15.817 align:middle line:90% think also in the 1960s. 01:04:15.817 --> 01:04:17.900 align:middle line:84% I'm pretty sure that these two people did not know 01:04:17.900 --> 01:04:21.020 align:middle line:90% what the other one was doing. 01:04:21.020 --> 01:04:23.620 align:middle line:84% And I guess it was probably also done in the setting of lines 01:04:23.620 --> 01:04:25.840 align:middle line:84% in finite fields in the '60s or '70s. 01:04:25.840 --> 01:04:27.118 align:middle line:90% But I don't know. 01:04:27.118 --> 01:04:28.160 align:middle line:90% I don't have a reference. 01:04:28.160 --> 01:04:30.638 align:middle line:90% 01:04:30.638 --> 01:04:32.180 align:middle line:84% And those people probably didn't know 01:04:32.180 --> 01:04:33.400 align:middle line:90% about either of these people. 01:04:33.400 --> 01:04:36.620 align:middle line:90% 01:04:36.620 --> 01:04:40.900 align:middle line:84% OK, so now, let me tell you what sieve theory is, 01:04:40.900 --> 01:04:43.480 align:middle line:84% and how it is analogous to what we've been doing. 01:04:43.480 --> 01:05:07.980 align:middle line:90% 01:05:07.980 --> 01:05:19.568 align:middle line:84% OK, so sieve theory takes place over the integers. 01:05:19.568 --> 01:05:22.110 align:middle line:84% And the main character in sieve theory is taking the integers 01:05:22.110 --> 01:05:26.905 align:middle line:84% and reducing it modulo q for different bases q. 01:05:26.905 --> 01:05:29.190 align:middle line:84% OK, so z, of course, is the integers. 01:05:29.190 --> 01:05:31.830 align:middle line:90% I'm going to write zq for-- 01:05:31.830 --> 01:05:34.070 align:middle line:90% some people write z mod qz. 01:05:34.070 --> 01:05:38.270 align:middle line:90% This is the integers modulo q. 01:05:38.270 --> 01:05:44.230 align:middle line:84% And there's a basic map pi sub q z to zq, which takes an integer 01:05:44.230 --> 01:05:45.850 align:middle line:90% and reduces it modulo q. 01:05:45.850 --> 01:05:51.630 align:middle line:90% 01:05:51.630 --> 01:05:55.670 align:middle line:84% OK, this map is going to play the role of our projections. 01:05:55.670 --> 01:05:58.550 align:middle line:84% So let me mention what is analogous. 01:05:58.550 --> 01:06:03.150 align:middle line:84% So we've been talking about maps pi theta that go from R2 to R, 01:06:03.150 --> 01:06:08.390 align:middle line:84% or one-dimensional subspace, or from Fq2 to Fq. 01:06:08.390 --> 01:06:11.230 align:middle line:90% 01:06:11.230 --> 01:06:13.800 align:middle line:84% And now, we're talking about the map pi cubed 01:06:13.800 --> 01:06:16.840 align:middle line:90% that goes from z to z mod q. 01:06:16.840 --> 01:06:18.800 align:middle line:84% What do all these things have in common? 01:06:18.800 --> 01:06:22.630 align:middle line:84% They are all group homomorphisms between abelian groups. 01:06:22.630 --> 01:06:35.040 align:middle line:90% 01:06:35.040 --> 01:06:45.720 align:middle line:84% OK, so what we're going to do in sieve theory 01:06:45.720 --> 01:06:49.120 align:middle line:84% is take a set, which is going to be now a set of integers, 01:06:49.120 --> 01:06:51.880 align:middle line:84% and think about all the different projections 01:06:51.880 --> 01:06:54.097 align:middle line:84% of its integer of the set, and think about 01:06:54.097 --> 01:06:55.680 align:middle line:84% how they're all related to each other, 01:06:55.680 --> 01:06:58.988 align:middle line:84% and how they're all related to the original set. 01:06:58.988 --> 01:07:00.780 align:middle line:84% So I have another little piece of notation. 01:07:00.780 --> 01:07:04.360 align:middle line:84% So this thing is the numbers from 1 up to n, 01:07:04.360 --> 01:07:07.640 align:middle line:84% which is a subset of the integers. 01:07:07.640 --> 01:07:10.633 align:middle line:84% And let me illustrate projection theory. 01:07:10.633 --> 01:07:13.050 align:middle line:84% Let me illustrate sieve theory with an interesting example 01:07:13.050 --> 01:07:16.590 align:middle line:84% of a set whose projections behave in a funny way. 01:07:16.590 --> 01:07:41.210 align:middle line:90% 01:07:41.210 --> 01:07:46.490 align:middle line:90% OK, so let's do an example. 01:07:46.490 --> 01:07:48.450 align:middle line:84% My example is going to be basically the set 01:07:48.450 --> 01:07:50.010 align:middle line:90% of square numbers. 01:07:50.010 --> 01:07:52.650 align:middle line:84% So it's going to be the set of little n squared, 01:07:52.650 --> 01:07:56.530 align:middle line:84% where little n goes from 1 up to big N to the 1/2. 01:07:56.530 --> 01:07:59.930 align:middle line:84% So it's a subset of the integers from 1 to N. 01:07:59.930 --> 01:08:05.770 align:middle line:84% And the cardinality is about the square root of N. 01:08:05.770 --> 01:08:11.860 align:middle line:84% What happens when I reduce this set modulo p? 01:08:11.860 --> 01:08:17.440 align:middle line:84% So if p is prime, when I take pi p of x, 01:08:17.440 --> 01:08:19.819 align:middle line:90% I get the quadratic residues. 01:08:19.819 --> 01:08:23.479 align:middle line:84% And the number of quadratic residues is p plus 1 over 2. 01:08:23.479 --> 01:08:26.380 align:middle line:90% So about half. 01:08:26.380 --> 01:08:30.140 align:middle line:84% So that's kind of an interesting and surprising fact. 01:08:30.140 --> 01:08:32.979 align:middle line:84% Of course, if p is bigger than square root of N, 01:08:32.979 --> 01:08:36.700 align:middle line:84% obviously, the size of this projection is smaller than x. 01:08:36.700 --> 01:08:38.859 align:middle line:84% So I might not even get all the quadratic residues 01:08:38.859 --> 01:08:42.600 align:middle line:84% if I take a big P. That's not surprising. 01:08:42.600 --> 01:08:46.597 align:middle line:84% But I could take p significantly less than the square root of N. 01:08:46.597 --> 01:08:50.040 align:middle line:84% And then when I do this projection, 01:08:50.040 --> 01:08:53.580 align:middle line:84% I only hit half of the elements in z mod p, 01:08:53.580 --> 01:08:57.006 align:middle line:84% and I hit each of those many times. 01:08:57.006 --> 01:08:59.800 align:middle line:84% That's not something that's likely to happen randomly. 01:08:59.800 --> 01:09:02.380 align:middle line:90% It's kind of striking. 01:09:02.380 --> 01:09:06.100 align:middle line:84% So one sample question from sieve theory 01:09:06.100 --> 01:09:13.069 align:middle line:84% is, how big could a set x be and have all of its projections 01:09:13.069 --> 01:09:15.029 align:middle line:90% be small like this? 01:09:15.029 --> 01:09:17.957 align:middle line:90% And what examples are there? 01:09:17.957 --> 01:09:19.499 align:middle line:84% How many sets like this can you find? 01:09:19.499 --> 01:09:22.069 align:middle line:90% 01:09:22.069 --> 01:09:23.670 align:middle line:90% All right. 01:09:23.670 --> 01:09:26.734 align:middle line:84% So using his ideas about sieve theory, 01:09:26.734 --> 01:09:28.109 align:middle line:84% one of the things that Linnik did 01:09:28.109 --> 01:09:31.170 align:middle line:84% was to answer some of this question. 01:09:31.170 --> 01:09:37.430 align:middle line:84% So theorem of Linnik in the 1940s says, 01:09:37.430 --> 01:09:40.883 align:middle line:84% if I have a subset of the integers from 1 to N, 01:09:40.883 --> 01:09:48.622 align:middle line:84% and if pi p of x is at most, p plus 1 over 2 for every P. 01:09:48.622 --> 01:09:50.330 align:middle line:84% So there's nothing so special about this. 01:09:50.330 --> 01:09:52.163 align:middle line:84% We could put other things here, but I'm just 01:09:52.163 --> 01:09:55.470 align:middle line:84% using this to compare with the squares. 01:09:55.470 --> 01:10:00.310 align:middle line:84% Then the conclusion is that the size of x is at most around N 01:10:00.310 --> 01:10:00.850 align:middle line:90% to the 1/2. 01:10:00.850 --> 01:10:05.710 align:middle line:90% 01:10:05.710 --> 01:10:07.800 align:middle line:84% So it says that the square numbers are just 01:10:07.800 --> 01:10:09.510 align:middle line:84% about the biggest set with this property. 01:10:09.510 --> 01:10:15.180 align:middle line:90% 01:10:15.180 --> 01:10:15.680 align:middle line:90% Cool. 01:10:15.680 --> 01:10:17.138 align:middle line:84% So we will give two proofs of this. 01:10:17.138 --> 01:10:18.940 align:middle line:84% We'll give a proof using double counting, 01:10:18.940 --> 01:10:20.940 align:middle line:84% and we'll give a proof using the Fourier method. 01:10:20.940 --> 01:10:24.640 align:middle line:90% 01:10:24.640 --> 01:10:27.460 align:middle line:84% There are lots of hard open problems in sieve theory. 01:10:27.460 --> 01:10:31.560 align:middle line:84% And one of them is to understand what the examples are. 01:10:31.560 --> 01:10:41.340 align:middle line:84% So the only known examples, only known examples 01:10:41.340 --> 01:10:47.600 align:middle line:84% where this theorem is roughly tight 01:10:47.600 --> 01:10:50.190 align:middle line:84% are the square numbers and close cousins. 01:10:50.190 --> 01:11:00.280 align:middle line:90% 01:11:00.280 --> 01:11:02.140 align:middle line:90% So sorry. 01:11:02.140 --> 01:11:02.740 align:middle line:90% It's messy. 01:11:02.740 --> 01:11:07.370 align:middle line:84% The square numbers, and close cousins of the square numbers. 01:11:07.370 --> 01:11:10.530 align:middle line:84% You could replace this with another degree 2 01:11:10.530 --> 01:11:12.250 align:middle line:90% polynomial in little n. 01:11:12.250 --> 01:11:14.468 align:middle line:84% And it's not hard to see it's kind of equivalent. 01:11:14.468 --> 01:11:15.760 align:middle line:90% And those are all the examples. 01:11:15.760 --> 01:11:19.530 align:middle line:90% 01:11:19.530 --> 01:11:23.090 align:middle line:84% OK, but people know basically nothing 01:11:23.090 --> 01:11:24.560 align:middle line:90% about classifying the examples. 01:11:24.560 --> 01:11:28.390 align:middle line:90% 01:11:28.390 --> 01:11:28.890 align:middle line:90% Cool. 01:11:28.890 --> 01:11:33.450 align:middle line:90% 01:11:33.450 --> 01:11:34.050 align:middle line:90% Cool. 01:11:34.050 --> 01:11:38.868 align:middle line:84% OK, so we'll give two proofs of this, 01:11:38.868 --> 01:11:40.410 align:middle line:84% one based on double counting, and one 01:11:40.410 --> 01:11:41.750 align:middle line:90% based on the Fourier method. 01:11:41.750 --> 01:11:43.570 align:middle line:84% Double counting, I think, we can do today. 01:11:43.570 --> 01:11:46.730 align:middle line:84% And the Fourier method, we'll do next time, next week. 01:11:46.730 --> 01:11:51.490 align:middle line:84% And first of all, I think that these results are pretty cool. 01:11:51.490 --> 01:11:54.330 align:middle line:84% But I also hope that you see that they're really extremely 01:11:54.330 --> 01:11:57.050 align:middle line:84% parallel to the results in geometric measure theory 01:11:57.050 --> 01:11:58.550 align:middle line:84% that we've just been talking about. 01:11:58.550 --> 01:12:03.280 align:middle line:90% 01:12:03.280 --> 01:12:03.780 align:middle line:90% OK. 01:12:03.780 --> 01:12:26.460 align:middle line:90% 01:12:26.460 --> 01:12:30.060 align:middle line:84% OK, let me make a little remark before we start. 01:12:30.060 --> 01:12:33.060 align:middle line:84% So I'm going to focus on projections mod p, 01:12:33.060 --> 01:12:34.580 align:middle line:90% where p is a prime. 01:12:34.580 --> 01:12:37.740 align:middle line:84% You could also reduce mod q, where q is not a prime. 01:12:37.740 --> 01:12:40.283 align:middle line:90% And all of the ideas still work. 01:12:40.283 --> 01:12:41.700 align:middle line:84% It's a little bit more complicated 01:12:41.700 --> 01:12:45.140 align:middle line:84% because if you take two numbers, q1 and q2 that share 01:12:45.140 --> 01:12:48.940 align:middle line:84% a common factor, then reducing mod q1 and reducing mod q2 01:12:48.940 --> 01:12:50.420 align:middle line:84% are kind of related to each other. 01:12:50.420 --> 01:12:51.920 align:middle line:84% And so if you write everything down, 01:12:51.920 --> 01:12:53.320 align:middle line:90% you have to keep track of that. 01:12:53.320 --> 01:12:55.660 align:middle line:84% That's just an extra thing to keep track of. 01:12:55.660 --> 01:12:58.878 align:middle line:84% And not so difficult, but it makes 01:12:58.878 --> 01:13:00.920 align:middle line:84% everything a little longer, and more complicated. 01:13:00.920 --> 01:13:04.010 align:middle line:84% And sometimes, you might want to do that, but in applications. 01:13:04.010 --> 01:13:09.170 align:middle line:84% But just to show the ideas, I'm going to focus on mod p. 01:13:09.170 --> 01:13:09.670 align:middle line:90% OK. 01:13:09.670 --> 01:13:31.830 align:middle line:90% 01:13:31.830 --> 01:13:33.470 align:middle line:90% All right. 01:13:33.470 --> 01:13:35.340 align:middle line:84% So here's the double counting method. 01:13:35.340 --> 01:13:42.870 align:middle line:90% 01:13:42.870 --> 01:13:44.700 align:middle line:90% So I'll call it theorem 1S. 01:13:44.700 --> 01:13:47.270 align:middle line:90% 01:13:47.270 --> 01:13:51.830 align:middle line:84% So it says if x is a subset of the numbers from 1 to N, 01:13:51.830 --> 01:14:05.000 align:middle line:84% and D is a set of primes less than or equal to N, 01:14:05.000 --> 01:14:08.920 align:middle line:84% and for every p in our set of primes, 01:14:08.920 --> 01:14:13.280 align:middle line:84% pi sub p of x is less than or equal to S, 01:14:13.280 --> 01:14:15.140 align:middle line:90% then we get a conclusion. 01:14:15.140 --> 01:14:20.240 align:middle line:90% 01:14:20.240 --> 01:14:27.400 align:middle line:84% Either x is bounded by 2S, or the number of directions is 01:14:27.400 --> 01:14:36.720 align:middle line:84% bounded roughly by S. So I guess this is most interesting 01:14:36.720 --> 01:14:40.800 align:middle line:84% if the number of directions is significantly bigger than S. 01:14:40.800 --> 01:14:45.920 align:middle line:84% So if we look at the squares, so suppose I know that pi p of x is 01:14:45.920 --> 01:14:49.020 align:middle line:90% bounded by p plus 1 over 2. 01:14:49.020 --> 01:14:56.600 align:middle line:84% So that's my S. And I need to do this-- 01:14:56.600 --> 01:14:57.816 align:middle line:90% yeah, let's see. 01:14:57.816 --> 01:15:04.490 align:middle line:90% 01:15:04.490 --> 01:15:06.190 align:middle line:84% Well, this may not quite work for this. 01:15:06.190 --> 01:15:10.530 align:middle line:90% 01:15:10.530 --> 01:15:12.312 align:middle line:90% Right. 01:15:12.312 --> 01:15:14.770 align:middle line:84% Actually, I'm not sure if this will imply Linnik's theorem. 01:15:14.770 --> 01:15:18.330 align:middle line:84% Linnik proved his theorem with the Fourier method. 01:15:18.330 --> 01:15:19.950 align:middle line:84% Let's come back to that next time. 01:15:19.950 --> 01:15:23.010 align:middle line:84% I think we have just enough time to do the nice double counting 01:15:23.010 --> 01:15:28.490 align:middle line:84% argument and then do the Fourier next time. 01:15:28.490 --> 01:15:34.810 align:middle line:84% OK, so proof, we're going to count coincidences. 01:15:34.810 --> 01:15:39.170 align:middle line:84% That's the set of x1, and x2, and x, 01:15:39.170 --> 01:15:43.710 align:middle line:84% and p in our set of directions, or a set of primes 01:15:43.710 --> 01:15:48.660 align:middle line:84% so that pi p of x1 equals pi p of x2. 01:15:48.660 --> 01:15:53.250 align:middle line:90% 01:15:53.250 --> 01:15:55.810 align:middle line:84% OK, so we'll count this two different ways. 01:15:55.810 --> 01:15:58.060 align:middle line:90% One is a lower bound. 01:15:58.060 --> 01:16:02.660 align:middle line:84% So star is at least for every direction, when 01:16:02.660 --> 01:16:05.660 align:middle line:84% I do this projection, there are only S images. 01:16:05.660 --> 01:16:09.300 align:middle line:84% So a typical image has x over S preimages. 01:16:09.300 --> 01:16:12.020 align:middle line:84% Square that for my choice of x1 and x2, 01:16:12.020 --> 01:16:14.580 align:middle line:84% and then multiply by S for the number 01:16:14.580 --> 01:16:16.980 align:middle line:90% of choices of the projection. 01:16:16.980 --> 01:16:23.100 align:middle line:90% So that's x squared D S inverse. 01:16:23.100 --> 01:16:24.670 align:middle line:90% OK, now what's our upper bound? 01:16:24.670 --> 01:16:28.460 align:middle line:90% 01:16:28.460 --> 01:16:31.240 align:middle line:84% So now, imagine fixing x1 and x2. 01:16:31.240 --> 01:16:33.920 align:middle line:84% And think about how many P's can satisfy this. 01:16:33.920 --> 01:16:38.540 align:middle line:90% 01:16:38.540 --> 01:16:50.020 align:middle line:84% OK, so if I have pi p of x1 is equal to pi p of x2, 01:16:50.020 --> 01:16:54.410 align:middle line:84% that tells us that p divides x2 minus x1. 01:16:54.410 --> 01:16:57.630 align:middle line:90% 01:16:57.630 --> 01:16:59.790 align:middle line:84% And there can't be that many different primes 01:16:59.790 --> 01:17:01.990 align:middle line:90% that divide x2 minus x1. 01:17:01.990 --> 01:17:05.270 align:middle line:90% x2 minus x1 has size n. 01:17:05.270 --> 01:17:08.910 align:middle line:84% So it could be, at most, log of N that do this. 01:17:08.910 --> 01:17:15.230 align:middle line:84% So star is upper bounded by, OK, so there 01:17:15.230 --> 01:17:17.750 align:middle line:90% is a case where x1 equals x2. 01:17:17.750 --> 01:17:22.070 align:middle line:84% So that gives me x times the number of directions. 01:17:22.070 --> 01:17:24.830 align:middle line:84% And then if x1 is not equal to x2, 01:17:24.830 --> 01:17:26.970 align:middle line:84% I have x squared choices for x1 and x2. 01:17:26.970 --> 01:17:30.530 align:middle line:84% But I have only log of N choices for the direction. 01:17:30.530 --> 01:17:39.750 align:middle line:90% 01:17:39.750 --> 01:17:42.370 align:middle line:84% OK, and now, I'll compare these to each other. 01:17:42.370 --> 01:17:56.520 align:middle line:90% 01:17:56.520 --> 01:18:03.440 align:middle line:84% OK, so I have x squared times D times S inverse is 01:18:03.440 --> 01:18:11.120 align:middle line:84% less than x times D plus x squared times log N. 01:18:11.120 --> 01:18:12.960 align:middle line:90% So there are two cases. 01:18:12.960 --> 01:18:17.720 align:middle line:84% If this one dominates, then I would have that x is bounded 01:18:17.720 --> 01:18:20.120 align:middle line:90% by 2S. 01:18:20.120 --> 01:18:24.720 align:middle line:84% Or if this one dominates, then the x's disappear, 01:18:24.720 --> 01:18:31.000 align:middle line:84% and I get that D is bounded by log of N times S. 01:18:31.000 --> 01:18:33.150 align:middle line:90% And that's the theorem. 01:18:33.150 --> 01:18:40.600 align:middle line:90% 01:18:40.600 --> 01:18:43.720 align:middle line:84% OK, so let me give you an example of this 01:18:43.720 --> 01:18:45.613 align:middle line:90% and argue that this is-- 01:18:45.613 --> 01:18:47.280 align:middle line:84% well, I'll give you an example, a flavor 01:18:47.280 --> 01:18:48.760 align:middle line:90% of what this is telling us. 01:18:48.760 --> 01:18:58.770 align:middle line:84% So suppose that pi p of x is bounded by N to the 2/3 for 10 01:18:58.770 --> 01:19:03.190 align:middle line:84% log n times n to the 2/3 different p. 01:19:03.190 --> 01:19:06.930 align:middle line:90% 01:19:06.930 --> 01:19:12.330 align:middle line:84% So if that's true, then the theorem tells us that x is 01:19:12.330 --> 01:19:13.740 align:middle line:90% bounded by N to the 2/3. 01:19:13.740 --> 01:19:23.934 align:middle line:90% 01:19:23.934 --> 01:19:27.130 align:middle line:84% So these primes might have size around this big, 01:19:27.130 --> 01:19:28.430 align:middle line:90% or a bit bigger. 01:19:28.430 --> 01:19:32.090 align:middle line:84% So these primes are much less than N. So each projection, 01:19:32.090 --> 01:19:34.730 align:middle line:84% the fact that each projection one at a time is small, 01:19:34.730 --> 01:19:37.220 align:middle line:84% it's not that convincing in terms of x being small. 01:19:37.220 --> 01:19:38.970 align:middle line:84% But knowing that all of these projections, 01:19:38.970 --> 01:19:41.190 align:middle line:84% or this large number of projections is small, 01:19:41.190 --> 01:19:44.000 align:middle line:84% it means that x is really that small. 01:19:44.000 --> 01:19:52.000 align:middle line:90%