WEBVTT 00:00:00.000 --> 00:00:01.905 align:middle line:90% [SQUEAKING] 00:00:01.905 --> 00:00:02.405 align:middle line:90% 00:00:02.405 --> 00:00:03.829 align:middle line:90% [RUSTLING] 00:00:03.829 --> 00:00:04.329 align:middle line:90% 00:00:04.329 --> 00:00:07.696 align:middle line:90% [CLICKING] 00:00:07.696 --> 00:00:12.040 align:middle line:90% 00:00:12.040 --> 00:00:16.340 align:middle line:84% PABLO SHMERKIN: So let's recall where we were last time. 00:00:16.340 --> 00:00:18.960 align:middle line:84% So we are trying to prove Bourgain's projection theorem. 00:00:18.960 --> 00:00:22.120 align:middle line:84% I guess I will state it again later. 00:00:22.120 --> 00:00:26.560 align:middle line:84% But last time, we hopefully accepted the following fact 00:00:26.560 --> 00:00:28.840 align:middle line:90% as true. 00:00:28.840 --> 00:00:45.280 align:middle line:84% OK, so maybe if a subset of 0, 1 is-- 00:00:45.280 --> 00:00:56.280 align:middle line:84% OK, so given S is 0, 1, there exists eta epsilon positive 00:00:56.280 --> 00:01:02.850 align:middle line:84% depending on S. And if we have a set, which is an S delta, 00:01:02.850 --> 00:01:07.690 align:middle line:84% delta to the minus eta set, then there 00:01:07.690 --> 00:01:12.810 align:middle line:84% exists a little A in the set such that the size of A plus 00:01:12.810 --> 00:01:18.730 align:middle line:84% little a, a grows by delta to the minus epsilon 00:01:18.730 --> 00:01:20.190 align:middle line:90% compared to the size of a. 00:01:20.190 --> 00:01:23.090 align:middle line:90% 00:01:23.090 --> 00:01:26.370 align:middle line:84% So Larry pointed out, OK, so initially, 00:01:26.370 --> 00:01:29.450 align:middle line:84% instead of this, we have some polynomial q of A, 00:01:29.450 --> 00:01:31.490 align:middle line:90% such as aA minus aA. 00:01:31.490 --> 00:01:34.650 align:middle line:84% So Larry pointed out to me that if you follow exactly what you 00:01:34.650 --> 00:01:38.450 align:middle line:84% did in the finite-field setting to go from q of a to this, 00:01:38.450 --> 00:01:40.770 align:middle line:84% you need to use the non-concentration condition 00:01:40.770 --> 00:01:42.690 align:middle line:90% that we are trying to prove. 00:01:42.690 --> 00:01:46.150 align:middle line:84% But it is possible to do it in a way, which is not circular. 00:01:46.150 --> 00:01:50.050 align:middle line:84% So OK, so I said, I was not going 00:01:50.050 --> 00:01:51.750 align:middle line:90% to cover all of the details. 00:01:51.750 --> 00:01:53.690 align:middle line:84% So using Plunnecke-Ruzsa-- so Plunnecke-Ruzsa 00:01:53.690 --> 00:01:57.130 align:middle line:84% is the main tool, and suitable double counting, 00:01:57.130 --> 00:02:00.220 align:middle line:84% one can really prove this without using something 00:02:00.220 --> 00:02:02.860 align:middle line:90% that we haven't proved yet. 00:02:02.860 --> 00:02:05.860 align:middle line:84% OK, and the goal is to improve this 00:02:05.860 --> 00:02:09.340 align:middle line:84% by showing that A plus aA satisfies 00:02:09.340 --> 00:02:11.060 align:middle line:84% a better non-concentration condition that 00:02:11.060 --> 00:02:12.900 align:middle line:90% will allow us to iterate. 00:02:12.900 --> 00:02:16.340 align:middle line:84% So the goal is to show that A plus aA actually 00:02:16.340 --> 00:02:20.660 align:middle line:84% contains because it is not true that it always is, but it 00:02:20.660 --> 00:02:28.660 align:middle line:84% contains a delta S plus epsilon delta to the minus 00:02:28.660 --> 00:02:35.380 align:middle line:84% maybe let's call it eta prime set. 00:02:35.380 --> 00:02:39.260 align:middle line:84% So the eta can get worse when one iterates. 00:02:39.260 --> 00:02:41.080 align:middle line:84% But one iterates finitely many times. 00:02:41.080 --> 00:02:43.340 align:middle line:90% So this is not serious. 00:02:43.340 --> 00:02:45.380 align:middle line:90% OK. 00:02:45.380 --> 00:02:47.460 align:middle line:90% OK, maybe for a different. 00:02:47.460 --> 00:02:49.130 align:middle line:84% So the epsilon is also going to change. 00:02:49.130 --> 00:02:59.230 align:middle line:90% 00:02:59.230 --> 00:03:06.310 align:middle line:84% OK, so I'm basically going back to what 00:03:06.310 --> 00:03:07.650 align:middle line:90% I did at the end of last time. 00:03:07.650 --> 00:03:08.810 align:middle line:90% But I was rushing a bit. 00:03:08.810 --> 00:03:12.310 align:middle line:84% So I'm going to be a bit more precise, hopefully. 00:03:12.310 --> 00:03:19.600 align:middle line:84% Suppose that A is delta S delta to the minus eta square set. 00:03:19.600 --> 00:03:24.270 align:middle line:90% 00:03:24.270 --> 00:03:27.020 align:middle line:84% Well, in particular, it is an S delta, delta to the minus-- 00:03:27.020 --> 00:03:30.430 align:middle line:90% 00:03:30.430 --> 00:03:33.190 align:middle line:90% so I guess delta goes first. 00:03:33.190 --> 00:03:35.350 align:middle line:84% In particular, it is a delta S delta 00:03:35.350 --> 00:03:37.850 align:middle line:84% to the minus eta set because eta is a very small number. 00:03:37.850 --> 00:03:39.510 align:middle line:90% But why did I square it? 00:03:39.510 --> 00:03:45.610 align:middle line:90% Because then, it is OK. 00:03:45.610 --> 00:03:48.310 align:middle line:84% And as we saw last time, if we take 00:03:48.310 --> 00:03:50.990 align:middle line:84% a subset of a which is uniform, delta m 00:03:50.990 --> 00:03:54.980 align:middle line:84% uniform for some good choice of delta, 00:03:54.980 --> 00:03:58.240 align:middle line:84% delta will have to be chosen small enough in terms of eta, 00:03:58.240 --> 00:03:59.090 align:middle line:90% but nothing else. 00:03:59.090 --> 00:04:01.840 align:middle line:90% 00:04:01.840 --> 00:04:11.350 align:middle line:84% Then, so we may assume let's call it delta uniform. 00:04:11.350 --> 00:04:14.160 align:middle line:90% 00:04:14.160 --> 00:04:19.079 align:middle line:90% Or delta depends on eta. 00:04:19.079 --> 00:04:21.060 align:middle line:84% So that basically, this doesn't change. 00:04:21.060 --> 00:04:23.680 align:middle line:90% Maybe one has to multiply by 2. 00:04:23.680 --> 00:04:27.120 align:middle line:84% But essentially, it doesn't change. 00:04:27.120 --> 00:04:30.840 align:middle line:84% Well then, this set, because it's uniform now, 00:04:30.840 --> 00:04:34.640 align:middle line:84% is also rho, is delta to the minus 00:04:34.640 --> 00:04:37.670 align:middle line:90% eta squared z for every who. 00:04:37.670 --> 00:04:46.240 align:middle line:90% 00:04:46.240 --> 00:04:49.440 align:middle line:84% And then this implies that it is also 00:04:49.440 --> 00:04:52.690 align:middle line:90% a row is rho to the minus eta. 00:04:52.690 --> 00:04:54.790 align:middle line:84% So the problem here is that now, we have rho here, 00:04:54.790 --> 00:04:56.850 align:middle line:90% but we still have delta here. 00:04:56.850 --> 00:05:00.210 align:middle line:84% But we replace eta by eta squared so 00:05:00.210 --> 00:05:03.290 align:middle line:90% that we can have also rho here. 00:05:03.290 --> 00:05:07.810 align:middle line:84% And this will be for every who between delta and delta 00:05:07.810 --> 00:05:13.808 align:middle line:90% to the eta, right? 00:05:13.808 --> 00:05:16.100 align:middle line:84% Because we started with delta to the minus eta squared. 00:05:16.100 --> 00:05:19.490 align:middle line:90% 00:05:19.490 --> 00:05:22.770 align:middle line:84% OK, we don't go all the way to 1, but almost. 00:05:22.770 --> 00:05:24.650 align:middle line:90% Eta is very, very small. 00:05:24.650 --> 00:05:27.230 align:middle line:90% So morally, this is almost 1. 00:05:27.230 --> 00:05:34.810 align:middle line:90% 00:05:34.810 --> 00:05:37.730 align:middle line:84% So from here to here, we use the uniformity. 00:05:37.730 --> 00:05:41.770 align:middle line:84% So one consequence of uniformity that I explained last time 00:05:41.770 --> 00:05:48.650 align:middle line:84% is that a uniform delta z is also a rho set for every rho. 00:05:48.650 --> 00:05:53.890 align:middle line:84% And this is just trivial inequality 00:05:53.890 --> 00:05:56.240 align:middle line:84% using that rho is at most delta to the eta. 00:05:56.240 --> 00:06:02.690 align:middle line:90% 00:06:02.690 --> 00:06:06.010 align:middle line:84% OK, so this means that we can apply what we've already 00:06:06.010 --> 00:06:09.570 align:middle line:84% know for every scale rho between delta and delta 00:06:09.570 --> 00:06:12.110 align:middle line:84% to the eta, because it satisfies the assumptions. 00:06:12.110 --> 00:06:17.250 align:middle line:90% 00:06:17.250 --> 00:06:19.330 align:middle line:84% And another thing that I explained last time 00:06:19.330 --> 00:06:22.050 align:middle line:84% is that even though a priori, we only 00:06:22.050 --> 00:06:27.250 align:middle line:84% know that there exists an A so that A plus aA grows, 00:06:27.250 --> 00:06:29.910 align:middle line:84% by applying that to the set of exceptions, 00:06:29.910 --> 00:06:32.650 align:middle line:84% we get that this is true for almost 00:06:32.650 --> 00:06:35.610 align:middle line:90% every A in a strong sense. 00:06:35.610 --> 00:06:40.220 align:middle line:84% So if we put all of these together, what we get is that-- 00:06:40.220 --> 00:06:44.770 align:middle line:90% 00:06:44.770 --> 00:06:47.810 align:middle line:84% OK, so maybe let's recall what I just said. 00:06:47.810 --> 00:06:54.700 align:middle line:84% So in fact, the set of A. OK, so let's 00:06:54.700 --> 00:06:59.020 align:middle line:84% work with some finite set of rhos, which are powers of delta 00:06:59.020 --> 00:06:59.940 align:middle line:90% again. 00:06:59.940 --> 00:07:04.340 align:middle line:84% So we have delta, delta squared, up to delta, 00:07:04.340 --> 00:07:08.740 align:middle line:90% to some maybe m prime. 00:07:08.740 --> 00:07:12.860 align:middle line:84% So that this is equal to delta to the eta. 00:07:12.860 --> 00:07:16.900 align:middle line:84% And recall that little delta is delta to the m. 00:07:16.900 --> 00:07:20.040 align:middle line:84% So m prime is less than m, but it is close to m. 00:07:20.040 --> 00:07:23.420 align:middle line:90% 00:07:23.420 --> 00:07:28.780 align:middle line:84% And again, this is a finite set of scales, but not really 00:07:28.780 --> 00:07:30.540 align:middle line:90% finite, because OK. 00:07:30.540 --> 00:07:33.140 align:middle line:84% It is finite, but it grows with delta. 00:07:33.140 --> 00:07:35.620 align:middle line:84% But it grows logarithmically with delta. 00:07:35.620 --> 00:07:41.560 align:middle line:84% So m is logarithm of little delta in base, big delta. 00:07:41.560 --> 00:07:42.980 align:middle line:90% Big delta is fixed. 00:07:42.980 --> 00:07:45.740 align:middle line:90% So it's logarithmic in delta. 00:07:45.740 --> 00:07:46.760 align:middle line:90% And m prime is smaller. 00:07:46.760 --> 00:07:48.950 align:middle line:84% So it's also logarithmic in delta. 00:07:48.950 --> 00:08:01.510 align:middle line:84% OK, so for each rho, we know that the set of A's 00:08:01.510 --> 00:08:05.950 align:middle line:84% so that A plus rho A doesn't grow. 00:08:05.950 --> 00:08:06.710 align:middle line:90% Sorry. 00:08:06.710 --> 00:08:11.310 align:middle line:84% little a, but measurable scale rho doesn't 00:08:11.310 --> 00:08:20.150 align:middle line:84% grow by rho to the minus epsilon has 00:08:20.150 --> 00:08:29.990 align:middle line:84% size at most rho to the eta times the size of A. 00:08:29.990 --> 00:08:31.090 align:middle line:90% And why is this true? 00:08:31.090 --> 00:08:37.789 align:middle line:84% Because if this was not true, then we 00:08:37.789 --> 00:08:45.230 align:middle line:84% could apply the statement above to the scale rho 00:08:45.230 --> 00:08:48.998 align:middle line:84% to the set of exceptions to this set, and get a contradiction. 00:08:48.998 --> 00:09:13.920 align:middle line:90% 00:09:13.920 --> 00:09:17.620 align:middle line:84% OK, and this is very small in terms of delta. 00:09:17.620 --> 00:09:20.830 align:middle line:84% So rho to the eta is, at most, delta to eta squared. 00:09:20.830 --> 00:09:32.760 align:middle line:90% 00:09:32.760 --> 00:09:33.280 align:middle line:90% Sorry. 00:09:33.280 --> 00:09:38.200 align:middle line:90% Here, OK. 00:09:38.200 --> 00:09:41.320 align:middle line:84% OK, so here, we are counting a set of A's. 00:09:41.320 --> 00:09:44.450 align:middle line:84% So the only way in which we can really count a set of A's is 00:09:44.450 --> 00:09:46.290 align:middle line:90% by measuring some scale. 00:09:46.290 --> 00:09:47.710 align:middle line:90% Otherwise, it could be infinite. 00:09:47.710 --> 00:09:49.240 align:middle line:84% So we always have to choose a scale. 00:09:49.240 --> 00:09:52.010 align:middle line:90% 00:09:52.010 --> 00:09:53.210 align:middle line:90% So scale rho. 00:09:53.210 --> 00:09:56.000 align:middle line:84% OK, rho to the eta is, at most, delta to the eta squared. 00:09:56.000 --> 00:10:00.130 align:middle line:90% 00:10:00.130 --> 00:10:01.220 align:middle line:90% And this means that-- 00:10:01.220 --> 00:10:11.210 align:middle line:90% 00:10:11.210 --> 00:10:11.830 align:middle line:90% sorry. 00:10:11.830 --> 00:10:13.970 align:middle line:90% This is wrong. 00:10:13.970 --> 00:10:15.390 align:middle line:90% Let's go back and fix it. 00:10:15.390 --> 00:10:17.390 align:middle line:90% So delta is too large. 00:10:17.390 --> 00:10:18.590 align:middle line:90% So delta is a constant. 00:10:18.590 --> 00:10:23.270 align:middle line:84% So I want to be away from 1, not away from 0, but away from 1. 00:10:23.270 --> 00:10:25.370 align:middle line:84% So the rhos that we consider are delta 00:10:25.370 --> 00:10:30.510 align:middle line:84% to the m, which is little delta, delta to the minus 1. 00:10:30.510 --> 00:10:35.630 align:middle line:84% And then we stop at delta to the eta. 00:10:35.630 --> 00:10:38.450 align:middle line:90% 00:10:38.450 --> 00:10:39.470 align:middle line:90% Sorry about that. 00:10:39.470 --> 00:10:43.420 align:middle line:90% 00:10:43.420 --> 00:10:48.620 align:middle line:84% OK, because we are only looking at logarithmically many values 00:10:48.620 --> 00:10:54.020 align:middle line:84% of delta, OK, one has to work a little bit more because here, 00:10:54.020 --> 00:10:56.260 align:middle line:84% we are measuring things at scale rho. 00:10:56.260 --> 00:11:00.943 align:middle line:90% But rho is far away from 1. 00:11:00.943 --> 00:11:02.860 align:middle line:84% And there are only logarithmically many values 00:11:02.860 --> 00:11:03.820 align:middle line:90% of rho. 00:11:03.820 --> 00:11:06.900 align:middle line:84% So putting all together, what we get 00:11:06.900 --> 00:11:31.140 align:middle line:84% is that for all A outside of a small set, maybe 00:11:31.140 --> 00:11:41.590 align:middle line:84% something like a set of size delta to the eta, I don't know, 00:11:41.590 --> 00:11:53.710 align:middle line:84% cubed times the size of A. We get that A plus little aA grows 00:11:53.710 --> 00:11:56.580 align:middle line:90% for every scale in this family. 00:11:56.580 --> 00:12:14.470 align:middle line:90% 00:12:14.470 --> 00:12:16.150 align:middle line:84% OK, and now, it looks like we are 00:12:16.150 --> 00:12:19.750 align:middle line:84% getting close because, well, we want 00:12:19.750 --> 00:12:29.630 align:middle line:84% to show that this contains delta S plus epsilon something set. 00:12:29.630 --> 00:12:32.590 align:middle line:84% And a necessary condition is that we have growth 00:12:32.590 --> 00:12:33.280 align:middle line:90% at every scale. 00:12:33.280 --> 00:12:36.950 align:middle line:90% 00:12:36.950 --> 00:12:43.440 align:middle line:84% So if you have a delta S plus epsilon something set, 00:12:43.440 --> 00:12:45.880 align:middle line:90% then this will be true. 00:12:45.880 --> 00:12:48.680 align:middle line:84% And if we knew that this set is uniform, 00:12:48.680 --> 00:12:51.615 align:middle line:84% we would be done, because for uniform set, 00:12:51.615 --> 00:12:53.240 align:middle line:84% the condition of growing at every scale 00:12:53.240 --> 00:12:55.600 align:middle line:90% is necessary and sufficient. 00:12:55.600 --> 00:12:58.960 align:middle line:84% But we don't know that this set is uniform. 00:12:58.960 --> 00:13:00.260 align:middle line:90% So we are not done yet. 00:13:00.260 --> 00:13:03.960 align:middle line:84% So we are getting closer, but we are not done yet. 00:13:03.960 --> 00:13:05.520 align:middle line:90% Any questions? 00:13:05.520 --> 00:13:09.920 align:middle line:84% So this is more or less where we were last time. 00:13:09.920 --> 00:13:13.720 align:middle line:84% OK, so in order to be able to finish, 00:13:13.720 --> 00:13:16.880 align:middle line:90% we still have to work a bit. 00:13:16.880 --> 00:13:21.840 align:middle line:84% And there was a claim, that I was explaining last time, 00:13:21.840 --> 00:13:24.520 align:middle line:90% that we can improve this. 00:13:24.520 --> 00:13:27.140 align:middle line:84% So here, we are projecting A times A. 00:13:27.140 --> 00:13:29.720 align:middle line:84% So this is pi A of A times A. So we 00:13:29.720 --> 00:13:33.080 align:middle line:84% can improve this to project an arbitrary dense subset 00:13:33.080 --> 00:13:41.410 align:middle line:84% of A times A. So claim, if g is contained in A times A. 00:13:41.410 --> 00:13:45.050 align:middle line:90% And it has size at scale delta. 00:13:45.050 --> 00:13:49.610 align:middle line:84% Let's say delta to the eta squared, or something like this, 00:13:49.610 --> 00:14:00.490 align:middle line:84% relative to A times A, then Pa of G already grows. 00:14:00.490 --> 00:14:07.630 align:middle line:84% And in fact, we will also want to apply it at many scales. 00:14:07.630 --> 00:14:08.570 align:middle line:90% Or maybe not. 00:14:08.570 --> 00:14:09.190 align:middle line:90% We'll see. 00:14:09.190 --> 00:14:10.790 align:middle line:84% OK, let's start with scale delta. 00:14:10.790 --> 00:14:13.330 align:middle line:84% Then we'll see what we actually need. 00:14:13.330 --> 00:14:16.150 align:middle line:84% Then this already grows, let's say by epsilon over 2. 00:14:16.150 --> 00:14:23.440 align:middle line:90% 00:14:23.440 --> 00:14:27.330 align:middle line:84% OK, and I guess at this stage, we 00:14:27.330 --> 00:14:30.690 align:middle line:84% claim that there exists such an A. 00:14:30.690 --> 00:14:32.730 align:middle line:84% But once again, if there exists an A, 00:14:32.730 --> 00:14:35.450 align:middle line:84% then it is true for nearly all A by applying 00:14:35.450 --> 00:14:38.620 align:middle line:84% the fact that there exists an A to the exceptional set. 00:14:38.620 --> 00:14:40.920 align:middle line:84% So we can play this game as many times as we want. 00:14:40.920 --> 00:14:43.380 align:middle line:84% And we want to play it many times. 00:14:43.380 --> 00:14:47.780 align:middle line:84% OK, so last time, we were in the middle of proving this. 00:14:47.780 --> 00:14:49.540 align:middle line:90% So let's start again. 00:14:49.540 --> 00:14:54.100 align:middle line:84% Assume that 1 is in A. This is just to avoid to-- 00:14:54.100 --> 00:14:56.260 align:middle line:84% OK, just for notational simplicity. 00:14:56.260 --> 00:15:01.020 align:middle line:90% It's not important. 00:15:01.020 --> 00:15:05.180 align:middle line:84% Well, if A equals 1 works, we are 00:15:05.180 --> 00:15:07.820 align:middle line:84% done because we are trying to show that some element of A 00:15:07.820 --> 00:15:08.400 align:middle line:90% works. 00:15:08.400 --> 00:15:12.780 align:middle line:84% So if the element 1 happens to work, we are happy. 00:15:12.780 --> 00:15:17.380 align:middle line:90% Otherwise, 1 doesn't work. 00:15:17.380 --> 00:15:19.060 align:middle line:84% And the fact that 1 doesn't work means 00:15:19.060 --> 00:15:33.100 align:middle line:84% that there exists G, which is dense in A times A, 00:15:33.100 --> 00:15:38.280 align:middle line:84% and such that pi 1 of G doesn't grow too much. 00:15:38.280 --> 00:15:49.980 align:middle line:90% 00:15:49.980 --> 00:15:51.940 align:middle line:90% And this, OK. 00:15:51.940 --> 00:15:54.100 align:middle line:84% So all of these together are exactly 00:15:54.100 --> 00:15:56.442 align:middle line:84% the assumptions of Balog-Szemeredi-Gowers 00:15:56.442 --> 00:15:57.900 align:middle line:84% or rather the delta covering number 00:15:57.900 --> 00:15:59.160 align:middle line:90% of Balog-Szemeredi-Gowers. 00:15:59.160 --> 00:16:01.980 align:middle line:84% But as I explained a couple of lectures ago, 00:16:01.980 --> 00:16:04.940 align:middle line:84% Balog-Szemeredi-Gowers works exactly the same 00:16:04.940 --> 00:16:06.190 align:middle line:90% for delta covering numbers. 00:16:06.190 --> 00:16:09.500 align:middle line:90% 00:16:09.500 --> 00:16:15.340 align:middle line:84% So maybe let's recall Balog-Szemeredi-Gowers 00:16:15.340 --> 00:16:27.420 align:middle line:84% for delta-covering numbers if A at scale delta has size 00:16:27.420 --> 00:16:34.350 align:middle line:84% N, and there exists G contained in A times 00:16:34.350 --> 00:16:43.510 align:middle line:84% A such that the size of G at scale delta is at least 1 over K 00:16:43.510 --> 00:16:46.950 align:middle line:84% times the size of A plus A at scale delta. 00:16:46.950 --> 00:16:55.470 align:middle line:84% And if we project in the direction 1, this G, 00:16:55.470 --> 00:17:00.750 align:middle line:84% this doesn't grow a lot compared to the size of A. If all of this 00:17:00.750 --> 00:17:05.949 align:middle line:84% is true, then there exists a set A prime contained in A, 00:17:05.949 --> 00:17:08.150 align:middle line:90% which is fairly dense. 00:17:08.150 --> 00:17:12.150 align:middle line:84% So the size of at scale delta is at least K 00:17:12.150 --> 00:17:18.310 align:middle line:84% to minus a constant times the size of A. 00:17:18.310 --> 00:17:20.340 align:middle line:90% And A prime has a small subset. 00:17:20.340 --> 00:17:28.950 align:middle line:90% 00:17:28.950 --> 00:17:32.560 align:middle line:84% The doubling constant for A prime is, at most, K 00:17:32.560 --> 00:17:36.150 align:middle line:84% to a constant power times the size of A prime. 00:17:36.150 --> 00:17:40.080 align:middle line:90% 00:17:40.080 --> 00:17:44.360 align:middle line:84% OK, this is Balog-Szemeredi-Gowers. 00:17:44.360 --> 00:17:47.840 align:middle line:84% Well, here, we can take well, N is just the delta 00:17:47.840 --> 00:17:50.560 align:middle line:84% covering number of A, and we can take 00:17:50.560 --> 00:17:55.840 align:middle line:84% K to be the minimum between eta squared, delta to the minus eta 00:17:55.840 --> 00:17:59.080 align:middle line:84% squared, and delta to the minus epsilon over 2, which is really 00:17:59.080 --> 00:18:00.380 align:middle line:90% delta to the minus eta squared. 00:18:00.380 --> 00:18:02.720 align:middle line:84% You should imagine that eta is smaller than epsilon, 00:18:02.720 --> 00:18:03.710 align:middle line:90% even before squaring. 00:18:03.710 --> 00:18:08.000 align:middle line:90% 00:18:08.000 --> 00:18:13.800 align:middle line:84% OK, so we apply by Balog-Szemeredi-Gowers 00:18:13.800 --> 00:18:23.120 align:middle line:84% and let A prime be the corresponding set. 00:18:23.120 --> 00:18:43.170 align:middle line:90% 00:18:43.170 --> 00:18:56.720 align:middle line:84% OK, and then A prime is dense in A, and has small subset. 00:18:56.720 --> 00:19:22.930 align:middle line:90% 00:19:22.930 --> 00:19:28.980 align:middle line:84% OK, so A prime is a dense subset of A. A 00:19:28.980 --> 00:19:31.680 align:middle line:90% is a delta S something set. 00:19:31.680 --> 00:19:34.320 align:middle line:84% So A prime is also a delta S something set, 00:19:34.320 --> 00:19:36.420 align:middle line:84% because we saw that subsets of delta A something 00:19:36.420 --> 00:19:39.400 align:middle line:84% sets or delta S something sets where the constant. 00:19:39.400 --> 00:19:42.500 align:middle line:84% So basically, you have to multiply the constant by this. 00:19:42.500 --> 00:19:47.500 align:middle line:84% But A was a delta S delta to the epsilon square set. 00:19:47.500 --> 00:19:50.460 align:middle line:84% So this is still a delta S delta to the minus 00:19:50.460 --> 00:19:53.380 align:middle line:90% O of epsilon squared set. 00:19:53.380 --> 00:19:56.140 align:middle line:84% So we can apply what we've know to A prime. 00:19:56.140 --> 00:20:04.980 align:middle line:84% So A prime is a delta S delta to the minus O of eta square set. 00:20:04.980 --> 00:20:07.940 align:middle line:90% So there exists A-- 00:20:07.940 --> 00:20:14.170 align:middle line:84% an A prime, sorry, such that A prime plus alpha A prime grows. 00:20:14.170 --> 00:20:22.300 align:middle line:90% 00:20:22.300 --> 00:20:27.070 align:middle line:84% By applying the fact from the beginning to A prime, 00:20:27.070 --> 00:20:28.950 align:middle line:84% it still satisfies the assumptions. 00:20:28.950 --> 00:20:31.780 align:middle line:84% We have lost a constant here, but this is harmless. 00:20:31.780 --> 00:20:36.590 align:middle line:90% 00:20:36.590 --> 00:20:37.890 align:middle line:90% Any questions so far? 00:20:37.890 --> 00:20:42.190 align:middle line:90% 00:20:42.190 --> 00:20:44.870 align:middle line:84% OK, and now, we apply something that you've seen 00:20:44.870 --> 00:20:46.130 align:middle line:90% in the finite-field setting. 00:20:46.130 --> 00:20:48.630 align:middle line:84% And again, this works exactly the same way 00:20:48.630 --> 00:20:50.643 align:middle line:90% for delta covering sets. 00:20:50.643 --> 00:20:52.310 align:middle line:84% What you saw in the finite-field setting 00:20:52.310 --> 00:20:56.890 align:middle line:84% is that if you have a set with the smallest subset. 00:20:56.890 --> 00:20:59.190 align:middle line:90% So small pi 1 projection. 00:20:59.190 --> 00:21:06.150 align:middle line:84% And large pi A projection, then in fact, the pi A projection 00:21:06.150 --> 00:21:10.110 align:middle line:84% remains large if we pass to a dense subset 00:21:10.110 --> 00:21:13.470 align:middle line:90% of A prime just times A prime. 00:21:13.470 --> 00:21:18.000 align:middle line:84% OK, so this is another recall from the finite-field story. 00:21:18.000 --> 00:21:21.550 align:middle line:90% 00:21:21.550 --> 00:21:26.420 align:middle line:84% If A prime plus A prime, let's say again, 00:21:26.420 --> 00:21:33.320 align:middle line:84% it grows by, at most, K. And A prime plus little A prime, 00:21:33.320 --> 00:21:37.260 align:middle line:84% let's say, so A prime has small doubling. 00:21:37.260 --> 00:21:43.920 align:middle line:84% But if you apply projection by A, it grows by something. 00:21:43.920 --> 00:21:50.160 align:middle line:84% Then in fact, this implies that for every G in A prime times 00:21:50.160 --> 00:22:01.480 align:middle line:84% A prime, which is dense, let's say dense with threshold 1 00:22:01.480 --> 00:22:09.880 align:middle line:84% over K, everything measured with delta covering numbers, 00:22:09.880 --> 00:22:11.730 align:middle line:90% the projection is still large. 00:22:11.730 --> 00:22:18.120 align:middle line:90% 00:22:18.120 --> 00:22:21.520 align:middle line:84% And I think it's K to minus some constant. 00:22:21.520 --> 00:22:23.080 align:middle line:84% And the L deal basically survives. 00:22:23.080 --> 00:22:37.130 align:middle line:90% 00:22:37.130 --> 00:22:39.550 align:middle line:84% Does this look like something you've seen before? 00:22:39.550 --> 00:22:43.650 align:middle line:90% 00:22:43.650 --> 00:22:47.570 align:middle line:84% OK, does it look like something that-- 00:22:47.570 --> 00:22:54.910 align:middle line:84% OK, well, go back to your notes, or otherwise, just believe me, 00:22:54.910 --> 00:22:57.283 align:middle line:90% this is true. 00:22:57.283 --> 00:22:59.450 align:middle line:84% Larry told me you've seen something like this, maybe 00:22:59.450 --> 00:23:00.692 align:middle line:90% with different letters. 00:23:00.692 --> 00:23:02.150 align:middle line:84% Very likely with different letters. 00:23:02.150 --> 00:23:05.110 align:middle line:84% But the point is have a set of small doubling, 00:23:05.110 --> 00:23:07.570 align:middle line:90% but large pi A projection. 00:23:07.570 --> 00:23:09.750 align:middle line:84% Then the pi A projection is large, 00:23:09.750 --> 00:23:11.260 align:middle line:84% even after passing through a subset. 00:23:11.260 --> 00:23:14.570 align:middle line:90% 00:23:14.570 --> 00:23:16.890 align:middle line:84% OK, so it looks like we've almost 00:23:16.890 --> 00:23:23.700 align:middle line:84% won because look at the claim, and look at what we have here. 00:23:23.700 --> 00:23:25.160 align:middle line:90% OK, have we really won? 00:23:25.160 --> 00:23:28.100 align:middle line:90% 00:23:28.100 --> 00:23:31.580 align:middle line:84% Not exactly, because the claim was for A, 00:23:31.580 --> 00:23:35.060 align:middle line:84% and we got it for A prime, which is a dense subset of A, 00:23:35.060 --> 00:23:41.260 align:middle line:84% but it is not A. So one has to work a bit to really win. 00:23:41.260 --> 00:23:45.700 align:middle line:84% But you will have to trust me that one can win from here. 00:23:45.700 --> 00:23:49.340 align:middle line:84% So the claim was that, what we obtain 00:23:49.340 --> 00:23:54.360 align:middle line:84% for A prime is true for A. And we got it for A prime, 00:23:54.360 --> 00:23:57.880 align:middle line:84% not for A. So we'll briefly explain how to get it for A. 00:23:57.880 --> 00:24:00.100 align:middle line:84% But I will not do all of the details, 00:24:00.100 --> 00:24:04.140 align:middle line:84% because otherwise, we'll be here forever. 00:24:04.140 --> 00:24:09.310 align:middle line:84% Maybe let's keep the objective inside. 00:24:09.310 --> 00:24:31.630 align:middle line:90% 00:24:31.630 --> 00:24:33.260 align:middle line:90% OK, we can upgrade. 00:24:33.260 --> 00:24:38.310 align:middle line:90% 00:24:38.310 --> 00:24:40.750 align:middle line:90% A prime to A. 00:24:40.750 --> 00:24:46.930 align:middle line:84% And the idea is, well, either A prime is A, in which case, 00:24:46.930 --> 00:24:48.390 align:middle line:90% we win. 00:24:48.390 --> 00:24:52.630 align:middle line:84% Or if A prime is not all of A, but it is almost all 00:24:52.630 --> 00:24:55.230 align:middle line:84% of A, in the sense that A minus A prime is really small, 00:24:55.230 --> 00:25:00.750 align:middle line:84% we still win because so if the difference between A and A prime 00:25:00.750 --> 00:25:03.990 align:middle line:84% is really small, you just apply trivial bounds on the very 00:25:03.990 --> 00:25:05.990 align:middle line:90% small part, and we still win. 00:25:05.990 --> 00:25:16.910 align:middle line:84% So if A minus A prime is very small, very small maybe, 00:25:16.910 --> 00:25:21.350 align:middle line:84% of size less than delta to the plus eta, something like this, 00:25:21.350 --> 00:25:25.310 align:middle line:84% times the size of A, we are fine. 00:25:25.310 --> 00:25:33.610 align:middle line:90% Otherwise, do the same. 00:25:33.610 --> 00:25:40.500 align:middle line:84% So apply the same argument to A minus A prime. 00:25:40.500 --> 00:25:44.150 align:middle line:90% 00:25:44.150 --> 00:25:47.550 align:middle line:84% If A prime is not very dense in A, then A minus A prime 00:25:47.550 --> 00:25:51.310 align:middle line:84% is still a delta S delta to the minus eta square set. 00:25:51.310 --> 00:25:55.070 align:middle line:84% And we can do the same and get a new A prime. 00:25:55.070 --> 00:26:00.310 align:middle line:84% So maybe the original A prime, we can call it A prime 1. 00:26:00.310 --> 00:26:04.030 align:middle line:84% And if we apply the same thing, we get an A prime 2. 00:26:04.030 --> 00:26:07.350 align:middle line:84% Now at the same time, I have to play the game that if something 00:26:07.350 --> 00:26:11.470 align:middle line:84% is true for 1A, it's true for nearly all A because we found 00:26:11.470 --> 00:26:14.670 align:middle line:84% that there exists an A, and there exists an A prime. 00:26:14.670 --> 00:26:17.240 align:middle line:84% But in fact, there are lots of A's. 00:26:17.240 --> 00:26:19.647 align:middle line:84% But the A prime could depend on A. 00:26:19.647 --> 00:26:21.230 align:middle line:84% So one has to be a little bit careful. 00:26:21.230 --> 00:26:24.864 align:middle line:90% 00:26:24.864 --> 00:26:26.880 align:middle line:90% But one can play this game. 00:26:26.880 --> 00:26:31.760 align:middle line:84% And eventually, one gets, so all of these sets 00:26:31.760 --> 00:26:35.390 align:middle line:84% have large size because they are dense in A prime. 00:26:35.390 --> 00:26:39.720 align:middle line:90% 00:26:39.720 --> 00:26:42.960 align:middle line:90% Or is it A prime is dense in A? 00:26:42.960 --> 00:26:48.560 align:middle line:84% So at some point, and they are all disjoint, 00:26:48.560 --> 00:26:52.040 align:middle line:84% they are all disjoint because we take away the previous ones 00:26:52.040 --> 00:26:53.880 align:middle line:90% when we apply this again. 00:26:53.880 --> 00:26:58.440 align:middle line:84% So at some point, we reach a step 00:26:58.440 --> 00:27:08.440 align:middle line:84% where we have exhausted almost all of A. Let's say something 00:27:08.440 --> 00:27:09.400 align:middle line:90% like this. 00:27:09.400 --> 00:27:11.400 align:middle line:90% And then we stop. 00:27:11.400 --> 00:27:15.890 align:middle line:84% OK, but I'm cheating here because-- 00:27:15.890 --> 00:27:16.870 align:middle line:90% and then we stop. 00:27:16.870 --> 00:27:20.370 align:middle line:84% And then basically, we win because if we have a set 00:27:20.370 --> 00:27:23.970 align:middle line:90% G which is dense in A times A-- 00:27:23.970 --> 00:27:26.150 align:middle line:84% OK, we still don't really win yet. 00:27:26.150 --> 00:27:28.970 align:middle line:84% But if I said G is dense in A times A, 00:27:28.970 --> 00:27:34.730 align:middle line:84% it will be dense in AI prime times AG prime, maybe 00:27:34.730 --> 00:27:36.410 align:middle line:90% a different prime. 00:27:36.410 --> 00:27:38.030 align:middle line:90% OK, so one still has to work. 00:27:38.030 --> 00:27:39.230 align:middle line:90% OK, plus some work. 00:27:39.230 --> 00:27:39.920 align:middle line:90% Plus more work. 00:27:39.920 --> 00:27:44.170 align:middle line:90% 00:27:44.170 --> 00:27:47.030 align:middle line:84% And I'm basically ignoring the little a here. 00:27:47.030 --> 00:27:49.810 align:middle line:84% But it's OK to ignore the little a because everything 00:27:49.810 --> 00:27:53.890 align:middle line:84% you prove from one little a is true for nearly all little a. 00:27:53.890 --> 00:27:56.023 align:middle line:84% Maybe forget what I said, but OK, this 00:27:56.023 --> 00:27:57.190 align:middle line:90% is one thing you have to do. 00:27:57.190 --> 00:28:00.510 align:middle line:90% Just iterate, so iterate this. 00:28:00.510 --> 00:28:02.530 align:middle line:90% Here, we get a dense A prime. 00:28:02.530 --> 00:28:04.990 align:middle line:84% We want to get all of A, not a dense A prime. 00:28:04.990 --> 00:28:09.170 align:middle line:84% We iterate, plus some double counting. 00:28:09.170 --> 00:28:11.600 align:middle line:90% We get that plane. 00:28:11.600 --> 00:28:17.420 align:middle line:90% 00:28:17.420 --> 00:28:20.620 align:middle line:90% Any questions? 00:28:20.620 --> 00:28:21.760 align:middle line:90% So one has to work here. 00:28:21.760 --> 00:28:23.920 align:middle line:90% So I haven't proved it. 00:28:23.920 --> 00:28:26.260 align:middle line:84% So I don't expect you to see the proof. 00:28:26.260 --> 00:28:28.560 align:middle line:84% So one thing you have to do in the proof is iterate. 00:28:28.560 --> 00:28:33.300 align:middle line:84% But you have to do more things, and double count carefully. 00:28:33.300 --> 00:28:34.800 align:middle line:90% But eventually, one gets that. 00:28:34.800 --> 00:28:38.760 align:middle line:84% So one is able to upgrade A prime to A. OK, 00:28:38.760 --> 00:28:41.380 align:middle line:90% so that claim is true. 00:28:41.380 --> 00:28:43.860 align:middle line:84% So that claim says that there exists an A. 00:28:43.860 --> 00:28:47.460 align:middle line:84% And once there is one A, there are many A's, nearly all A's, 00:28:47.460 --> 00:28:50.840 align:middle line:84% so that the pi A projection of any dense subset of A plus A 00:28:50.840 --> 00:28:51.340 align:middle line:90% grows. 00:28:51.340 --> 00:28:56.460 align:middle line:90% 00:28:56.460 --> 00:29:02.090 align:middle line:84% And now, we are really close to the end of that goal. 00:29:02.090 --> 00:29:06.140 align:middle line:90% 00:29:06.140 --> 00:29:13.930 align:middle line:84% So A plus little aA is not necessarily uniform, 00:29:13.930 --> 00:29:17.270 align:middle line:84% but we know that it contains a uniform subset. 00:29:17.270 --> 00:29:19.430 align:middle line:84% And when we pass to a uniform subset, 00:29:19.430 --> 00:29:26.830 align:middle line:84% we can make it large for any subadditive function, set 00:29:26.830 --> 00:29:29.110 align:middle line:90% function mu. 00:29:29.110 --> 00:29:32.070 align:middle line:84% OK, so this is the mu we are going to use now. 00:29:32.070 --> 00:29:35.810 align:middle line:90% So mu of B is going to be, OK. 00:29:35.810 --> 00:29:39.980 align:middle line:84% So fix little a for which the claim holds. 00:29:39.980 --> 00:29:52.710 align:middle line:90% 00:29:52.710 --> 00:29:54.630 align:middle line:90% So this is the claim star. 00:29:54.630 --> 00:29:59.070 align:middle line:84% So we fix a little a so that this holds. 00:29:59.070 --> 00:30:03.510 align:middle line:84% And then mu of B, and here, B is a subset of 0,1, 00:30:03.510 --> 00:30:08.870 align:middle line:84% so it's a subset of A really, is the delta covering number not 00:30:08.870 --> 00:30:23.070 align:middle line:90% of B, but of pi A inverse of B. 00:30:23.070 --> 00:30:24.320 align:middle line:90% So this is subadditive, right? 00:30:24.320 --> 00:30:28.800 align:middle line:84% Because what is mu of B1 union B2? 00:30:28.800 --> 00:30:33.240 align:middle line:84% Well, if you take primitives, you 00:30:33.240 --> 00:30:37.140 align:middle line:84% get something which is contained in the union of the primitives. 00:30:37.140 --> 00:30:39.620 align:middle line:84% And then the delta covering number is subadditive. 00:30:39.620 --> 00:30:44.200 align:middle line:84% AUDIENCE: Isn't pi inverse of E intersecting cross A? 00:30:44.200 --> 00:30:45.033 align:middle line:90% PABLO SHMERKIN: Yes. 00:30:45.033 --> 00:30:46.200 align:middle line:90% Sorry. 00:30:46.200 --> 00:30:46.900 align:middle line:90% Thank you. 00:30:46.900 --> 00:30:57.440 align:middle line:90% 00:30:57.440 --> 00:31:01.425 align:middle line:84% So this is finitely subadditive, because delta covering number 00:31:01.425 --> 00:31:02.050 align:middle line:90% is subadditive. 00:31:02.050 --> 00:31:04.840 align:middle line:84% And we are just taking delta covering number together 00:31:04.840 --> 00:31:07.580 align:middle line:84% with the preimage, which is really subadditive. 00:31:07.580 --> 00:31:18.130 align:middle line:90% 00:31:18.130 --> 00:31:23.610 align:middle line:84% OK, so by lemma from last time, by the uniformization lemma 00:31:23.610 --> 00:31:37.210 align:middle line:84% from last time, there exists on A prime 00:31:37.210 --> 00:31:46.930 align:middle line:84% contained in A, which is uniform, sorry, not in A. Ah. 00:31:46.930 --> 00:31:52.450 align:middle line:84% There exists in B contained in a plus little aA. 00:31:52.450 --> 00:31:54.090 align:middle line:84% So we are going to apply uniformization 00:31:54.090 --> 00:31:55.510 align:middle line:90% to a plus little aA. 00:31:55.510 --> 00:31:59.330 align:middle line:90% 00:31:59.330 --> 00:32:02.290 align:middle line:84% We are trying to show that this set contains a delta S 00:32:02.290 --> 00:32:03.650 align:middle line:90% plus epsilon set. 00:32:03.650 --> 00:32:06.180 align:middle line:90% So this is our ultimate goal. 00:32:06.180 --> 00:32:09.220 align:middle line:84% And we know that if this set was uniform, 00:32:09.220 --> 00:32:12.140 align:middle line:84% it would be enough to show that it grows at every scale 00:32:12.140 --> 00:32:13.940 align:middle line:90% to reach the conclusion. 00:32:13.940 --> 00:32:16.540 align:middle line:84% So we want to make it uniform, but we 00:32:16.540 --> 00:32:21.020 align:middle line:84% need to know that the uniform subset grows at every scale. 00:32:21.020 --> 00:32:24.700 align:middle line:84% OK, so by uniformization, there exists a B 00:32:24.700 --> 00:32:30.900 align:middle line:84% in A plus little aA, such that this mu of B 00:32:30.900 --> 00:32:36.580 align:middle line:84% is larger than delta to the minus eta squared 00:32:36.580 --> 00:32:40.420 align:middle line:90% times the measure of A plus aA. 00:32:40.420 --> 00:32:42.500 align:middle line:84% And what is the measure of A plus aA? 00:32:42.500 --> 00:32:55.380 align:middle line:84% This is just the delta covering number of A times A. 00:32:55.380 --> 00:33:06.110 align:middle line:84% So this is the delta covering number of pi A inverse of B. So 00:33:06.110 --> 00:33:07.110 align:middle line:90% this is a-- 00:33:07.110 --> 00:33:07.990 align:middle line:90% sorry. 00:33:07.990 --> 00:33:13.670 align:middle line:84% Again, I forgot to intersect with A times A. 00:33:13.670 --> 00:33:17.440 align:middle line:84% So this is the G to which we can apply the claim. 00:33:17.440 --> 00:33:20.070 align:middle line:90% 00:33:20.070 --> 00:33:24.230 align:middle line:84% So this is a dense subset of A times A. 00:33:24.230 --> 00:33:28.070 align:middle line:84% And we know that for every dense subset of A times A, the pi 00:33:28.070 --> 00:33:30.830 align:middle line:90% A projection grows. 00:33:30.830 --> 00:33:34.310 align:middle line:84% In particular, the pi A projection of this grows. 00:33:34.310 --> 00:33:35.850 align:middle line:90% But this is by A inverse. 00:33:35.850 --> 00:33:38.630 align:middle line:84% So if we apply pi A again, we land in B. 00:33:38.630 --> 00:33:41.537 align:middle line:84% And in particular, we land in A plus little aA. 00:33:41.537 --> 00:33:43.620 align:middle line:84% But in particular, we land in B, which is uniform. 00:33:43.620 --> 00:34:05.530 align:middle line:90% 00:34:05.530 --> 00:34:14.199 align:middle line:84% OK, so I apply claim star to G equals 00:34:14.199 --> 00:34:19.480 align:middle line:84% pi I inverse of B intersection A times A, 00:34:19.480 --> 00:34:21.980 align:middle line:84% that satisfies the density assumption. 00:34:21.980 --> 00:34:25.360 align:middle line:90% 00:34:25.360 --> 00:34:28.679 align:middle line:84% And this is because uniform sets can be taken dense. 00:34:28.679 --> 00:34:31.480 align:middle line:84% So again here, I have to choose maybe a different delta 00:34:31.480 --> 00:34:33.920 align:middle line:84% than before, although I guess the same delta works 00:34:33.920 --> 00:34:36.080 align:middle line:84% because it's the same numerology. 00:34:36.080 --> 00:34:39.440 align:middle line:90% So this is true. 00:34:39.440 --> 00:34:41.639 align:middle line:84% We can take uniform sets, which are 00:34:41.639 --> 00:34:44.880 align:middle line:84% as dense in the exponential sense as we want. 00:34:44.880 --> 00:34:48.320 align:middle line:84% This is exactly what's going on here. 00:34:48.320 --> 00:34:57.424 align:middle line:84% OK, and the conclusion is that the delta covering number grows. 00:34:57.424 --> 00:35:09.060 align:middle line:90% 00:35:09.060 --> 00:35:09.560 align:middle line:90% Sorry. 00:35:09.560 --> 00:35:11.140 align:middle line:90% Not pi A of B. Ah. 00:35:11.140 --> 00:35:17.040 align:middle line:90% 00:35:17.040 --> 00:35:23.670 align:middle line:84% Pi A of A times A intersected with B, I guess. 00:35:23.670 --> 00:35:57.120 align:middle line:90% 00:35:57.120 --> 00:36:03.290 align:middle line:84% OK, so we are one step closer because now, 00:36:03.290 --> 00:36:06.280 align:middle line:84% we have a uniform subset of A times aA. 00:36:06.280 --> 00:36:09.490 align:middle line:90% 00:36:09.490 --> 00:36:10.980 align:middle line:90% And the delta covering number-- 00:36:10.980 --> 00:36:18.130 align:middle line:90% 00:36:18.130 --> 00:36:20.570 align:middle line:84% well, I guess B is a subset of these. 00:36:20.570 --> 00:36:24.010 align:middle line:84% So here, one will have just B. So the delta-covering number 00:36:24.010 --> 00:36:25.890 align:middle line:90% grows. 00:36:25.890 --> 00:36:27.530 align:middle line:84% We know that if a set is uniform, 00:36:27.530 --> 00:36:31.650 align:middle line:84% and the rho covering number grows for every rho, 00:36:31.650 --> 00:36:33.970 align:middle line:84% and in fact, we don't need every rho, 00:36:33.970 --> 00:36:37.530 align:middle line:84% it's enough to consider rhos, which are powers of delta. 00:36:37.530 --> 00:36:46.090 align:middle line:84% Then they are delta S plus the growth set. 00:36:46.090 --> 00:36:47.370 align:middle line:90% So we just repeat. 00:36:47.370 --> 00:36:50.610 align:middle line:84% So here, we have something for delta. 00:36:50.610 --> 00:36:54.290 align:middle line:90% But then, OK. 00:36:54.290 --> 00:36:56.230 align:middle line:84% Let's see what is the right order to do this. 00:36:56.230 --> 00:36:58.980 align:middle line:84% So I want to claim that what we did for delta, 00:36:58.980 --> 00:37:00.420 align:middle line:90% we can do for any scale. 00:37:00.420 --> 00:37:02.700 align:middle line:84% So there is nothing special about delta. 00:37:02.700 --> 00:37:06.660 align:middle line:84% Because A itself is uniform, all of these that we did for delta, 00:37:06.660 --> 00:37:11.020 align:middle line:84% we can do it for any other scale rho. 00:37:11.020 --> 00:37:16.150 align:middle line:84% But the uniform set that we get can depend on the scale. 00:37:16.150 --> 00:37:24.340 align:middle line:90% 00:37:24.340 --> 00:37:27.060 align:middle line:84% So what is the right order to do this? 00:37:27.060 --> 00:37:29.740 align:middle line:84% So the goal is to use the lemma from last time that 00:37:29.740 --> 00:37:33.060 align:middle line:84% says that if it is uniform, and the rho-covering number is 00:37:33.060 --> 00:37:35.740 align:middle line:84% large for every row, and in fact, one doesn't need 00:37:35.740 --> 00:37:37.780 align:middle line:84% every, every rho, it's enough to consider powers 00:37:37.780 --> 00:37:40.500 align:middle line:84% of the delta base in the uniformity, 00:37:40.500 --> 00:37:44.540 align:middle line:84% then it is a delta S set, where the S comes 00:37:44.540 --> 00:37:48.186 align:middle line:84% from the size of the rho-covering numbers. 00:37:48.186 --> 00:37:54.740 align:middle line:84% And OK, by using this idea for every rho, 00:37:54.740 --> 00:37:57.350 align:middle line:84% so we did it for delta, but the assumption 00:37:57.350 --> 00:37:59.230 align:middle line:84% is called for every rho, at least 00:37:59.230 --> 00:38:01.310 align:middle line:84% for every rho which is not very close to 1. 00:38:01.310 --> 00:38:02.340 align:middle line:90% So we can do the same. 00:38:02.340 --> 00:38:05.550 align:middle line:90% 00:38:05.550 --> 00:38:08.158 align:middle line:84% The only danger is that we could get a different B 00:38:08.158 --> 00:38:09.610 align:middle line:90% for different rhos. 00:38:09.610 --> 00:38:10.760 align:middle line:90% We could get different B's. 00:38:10.760 --> 00:38:30.270 align:middle line:90% 00:38:30.270 --> 00:38:31.830 align:middle line:90% I'm not very happy about this. 00:38:31.830 --> 00:38:34.890 align:middle line:90% So how do we deal with this? 00:38:34.890 --> 00:38:37.750 align:middle line:90% 00:38:37.750 --> 00:38:41.750 align:middle line:84% One thing one could try to do, but I think we lose too much, 00:38:41.750 --> 00:38:45.950 align:middle line:90% is OK, delta is delta to the m. 00:38:45.950 --> 00:38:48.910 align:middle line:84% Now we want to go to delta to the m minus 1. 00:38:48.910 --> 00:38:54.040 align:middle line:84% So we could just replace A plus aA by B. 00:38:54.040 --> 00:38:56.500 align:middle line:84% But then we will have another subset of B, 00:38:56.500 --> 00:38:58.640 align:middle line:84% and we are going to use this factor logarithmically 00:38:58.640 --> 00:38:59.680 align:middle line:90% many times. 00:38:59.680 --> 00:39:01.540 align:middle line:90% And that is true many times. 00:39:01.540 --> 00:39:02.040 align:middle line:90% So yeah. 00:39:02.040 --> 00:39:03.112 align:middle line:90% I don't want to do that. 00:39:03.112 --> 00:39:04.070 align:middle line:90% It would lose too much. 00:39:04.070 --> 00:39:08.520 align:middle line:90% 00:39:08.520 --> 00:39:10.840 align:middle line:84% OK, so there is something I'm missing. 00:39:10.840 --> 00:39:13.600 align:middle line:84% And I'm not going to figure it out right now. 00:39:13.600 --> 00:39:17.480 align:middle line:84% So because I told you that-- yes. 00:39:17.480 --> 00:39:19.840 align:middle line:84% AUDIENCE: Can you prove the claim simultaneously 00:39:19.840 --> 00:39:23.400 align:middle line:84% for all scales, replace that delta by rhos, 00:39:23.400 --> 00:39:24.660 align:middle line:90% all the different rhos? 00:39:24.660 --> 00:39:28.657 align:middle line:84% And then I think that would fix it. 00:39:28.657 --> 00:39:30.240 align:middle line:84% PABLO SHMERKIN: Yeah, you want to have 00:39:30.240 --> 00:39:31.690 align:middle line:90% the same G for every scale? 00:39:31.690 --> 00:39:32.440 align:middle line:90% AUDIENCE: Yes. 00:39:32.440 --> 00:39:33.320 align:middle line:90% PABLO SHMERKIN: Yeah. 00:39:33.320 --> 00:39:35.570 align:middle line:84% AUDIENCE: Like if you could prove that stronger claim, 00:39:35.570 --> 00:39:37.967 align:middle line:90% then it will solve it. 00:39:37.967 --> 00:39:38.800 align:middle line:90% PABLO SHMERKIN: Yes. 00:39:38.800 --> 00:39:43.520 align:middle line:84% But why can't we get the same G for all the scales? 00:39:43.520 --> 00:39:46.360 align:middle line:84% OK, because I told you I wouldn't give a complete proof, 00:39:46.360 --> 00:39:48.220 align:middle line:84% and just a sketch of some of the ideas, 00:39:48.220 --> 00:39:50.640 align:middle line:84% but here is a sketch of some of the ideas. 00:39:50.640 --> 00:39:52.680 align:middle line:84% And it's something technical that-- 00:39:52.680 --> 00:39:55.730 align:middle line:84% so really, the ideas are what I explained. 00:39:55.730 --> 00:39:58.330 align:middle line:84% And we're missing something technical. 00:39:58.330 --> 00:40:01.150 align:middle line:90% It's not fundamental. 00:40:01.150 --> 00:40:06.610 align:middle line:84% I would say just to summarize what's been going on, 00:40:06.610 --> 00:40:11.270 align:middle line:84% I still have to explain once we prove that goal, 00:40:11.270 --> 00:40:15.550 align:middle line:84% how to continue with the rest of Bourgain's original theorem. 00:40:15.550 --> 00:40:17.930 align:middle line:84% But this is really the most difficult step 00:40:17.930 --> 00:40:21.370 align:middle line:84% where things really change from the finite-field setting. 00:40:21.370 --> 00:40:26.170 align:middle line:84% OK, so let me sort of briefly explain, 00:40:26.170 --> 00:40:28.970 align:middle line:84% again, everything that's been going on. 00:40:28.970 --> 00:40:31.270 align:middle line:90% So we know that A plus aA grows. 00:40:31.270 --> 00:40:34.130 align:middle line:84% But we want to show that it also satisfies a stronger 00:40:34.130 --> 00:40:36.370 align:middle line:84% non-concentration assumption, that the S 00:40:36.370 --> 00:40:39.930 align:middle line:84% in the non-concentration assumption also grows. 00:40:39.930 --> 00:40:42.670 align:middle line:84% OK, first, we can take A to be uniform. 00:40:42.670 --> 00:40:46.170 align:middle line:84% If A is uniform, then we know that A itself 00:40:46.170 --> 00:40:48.930 align:middle line:90% is a rho S set for every rho. 00:40:48.930 --> 00:40:51.220 align:middle line:84% And this allows us to reach conclusions 00:40:51.220 --> 00:40:54.900 align:middle line:84% for every scale rho, which is something that we have to use. 00:40:54.900 --> 00:40:57.160 align:middle line:84% In the step that I'm missing as well. 00:40:57.160 --> 00:41:01.660 align:middle line:84% We used it before, but clearly, we have to use it again. 00:41:01.660 --> 00:41:06.500 align:middle line:84% And then we know that A plus little aA grows at every scale. 00:41:06.500 --> 00:41:11.060 align:middle line:84% If it was true that a plus little aA was uniform, 00:41:11.060 --> 00:41:13.280 align:middle line:84% then we would win, because for uniform sets, 00:41:13.280 --> 00:41:15.960 align:middle line:84% we saw last time that if they are large at every scale, 00:41:15.960 --> 00:41:18.700 align:middle line:84% then they satisfy the corresponding non-concentration 00:41:18.700 --> 00:41:21.940 align:middle line:90% condition. 00:41:21.940 --> 00:41:28.220 align:middle line:84% So we have to take a large uniform subset of A, 00:41:28.220 --> 00:41:30.040 align:middle line:90% but large with respect to what? 00:41:30.040 --> 00:41:31.860 align:middle line:84% Well, large with respect to this. 00:41:31.860 --> 00:41:37.340 align:middle line:84% Because this is what allow us to use the fact that if we project 00:41:37.340 --> 00:41:40.533 align:middle line:84% something dense in A times A, then we grow. 00:41:40.533 --> 00:41:42.700 align:middle line:84% And how do we know that if we inject something dense 00:41:42.700 --> 00:41:44.940 align:middle line:90% in A times A, then we grow? 00:41:44.940 --> 00:41:46.640 align:middle line:90% We have to combine two things. 00:41:46.640 --> 00:41:48.260 align:middle line:84% The first Balog-Szemeredi-Gowers. 00:41:48.260 --> 00:41:50.770 align:middle line:90% 00:41:50.770 --> 00:41:52.390 align:middle line:84% So for Balog-Szemeredi-Gowers we just 00:41:52.390 --> 00:41:55.970 align:middle line:84% take an arbitrary element of A, in this case, we took 1, 00:41:55.970 --> 00:41:58.870 align:middle line:84% but it can be an arbitrary element of A, 00:41:58.870 --> 00:42:00.710 align:middle line:90% we apply Balog-Szemeredi-Gowers. 00:42:00.710 --> 00:42:02.630 align:middle line:84% So either that element already works 00:42:02.630 --> 00:42:06.690 align:middle line:84% and we are done for the claim star, or if it doesn't work, 00:42:06.690 --> 00:42:10.070 align:middle line:84% then we are under the assumption of Balog-Szemeredi-Gowers. 00:42:10.070 --> 00:42:13.190 align:middle line:84% We apply Balog-Szemeredi-Gowers, and then we 00:42:13.190 --> 00:42:14.830 align:middle line:84% end up with the set that satisfies 00:42:14.830 --> 00:42:16.750 align:middle line:84% the assumption of something else that you've 00:42:16.750 --> 00:42:18.510 align:middle line:84% seen in the finite-field setting, which 00:42:18.510 --> 00:42:20.270 align:middle line:84% is that if you have a very small subset, 00:42:20.270 --> 00:42:23.550 align:middle line:84% an expansion under pi A, then this expansion under pi A 00:42:23.550 --> 00:42:25.550 align:middle line:84% is robust under pass into subsets. 00:42:25.550 --> 00:42:30.230 align:middle line:84% So in either case, we get that pi A of dense subset of A times 00:42:30.230 --> 00:42:31.590 align:middle line:90% A is large. 00:42:31.590 --> 00:42:36.590 align:middle line:84% And this allows us to use uniformity with this mu. 00:42:36.590 --> 00:42:43.350 align:middle line:84% So that's a bit of a summary of the main steps to show the goal. 00:42:43.350 --> 00:42:47.190 align:middle line:90% OK, plus technical details. 00:42:47.190 --> 00:42:49.620 align:middle line:84% I can recall right now, sorry about that. 00:42:49.620 --> 00:42:55.200 align:middle line:90% 00:42:55.200 --> 00:42:58.080 align:middle line:90% This implies the goal. 00:42:58.080 --> 00:42:58.840 align:middle line:90% The goal? 00:42:58.840 --> 00:42:59.350 align:middle line:90% That goal. 00:42:59.350 --> 00:43:06.080 align:middle line:90% 00:43:06.080 --> 00:43:09.200 align:middle line:84% OK, why did you spend so much time towards this goal? 00:43:09.200 --> 00:43:11.600 align:middle line:90% Because now, we can iterate. 00:43:11.600 --> 00:43:15.980 align:middle line:84% We knew that if A is a delta S delta to the minus epsilon set, 00:43:15.980 --> 00:43:17.780 align:middle line:90% then A plus A grows. 00:43:17.780 --> 00:43:19.480 align:middle line:84% But not only it grows, it contains 00:43:19.480 --> 00:43:23.860 align:middle line:84% the delta S plus epsilon delta to the minus eta prime set. 00:43:23.860 --> 00:43:26.080 align:middle line:90% And then we can iterate. 00:43:26.080 --> 00:43:35.280 align:middle line:84% So A plus little aA is our new A. 00:43:35.280 --> 00:43:49.250 align:middle line:84% So iterating A goes to A plus aA for every epsilon-- 00:43:49.250 --> 00:43:50.670 align:middle line:90% Maybe let's not call it epsilon. 00:43:50.670 --> 00:43:57.250 align:middle line:84% For every tau, there exists a polynomial that depends on tau. 00:43:57.250 --> 00:44:02.530 align:middle line:84% It's a very large degree if tau is close to 0 00:44:02.530 --> 00:44:09.690 align:middle line:84% such that Q tau of A is a delta 1 minus tau. 00:44:09.690 --> 00:44:15.210 align:middle line:84% So the S can become arbitrarily close to 1. 00:44:15.210 --> 00:44:18.210 align:middle line:84% Delta to the minus some eta tilde, 00:44:18.210 --> 00:44:22.290 align:middle line:84% that depends on how many times you have to iterate set. 00:44:22.290 --> 00:44:23.880 align:middle line:90% And by is, I mean, contains. 00:44:23.880 --> 00:44:28.130 align:middle line:90% 00:44:28.130 --> 00:44:29.690 align:middle line:90% Yes. 00:44:29.690 --> 00:44:31.290 align:middle line:84% AUDIENCE: You said that the epsilon 00:44:31.290 --> 00:44:36.048 align:middle line:90% is going to change [INAUDIBLE]. 00:44:36.048 --> 00:44:38.090 align:middle line:84% PABLO SHMERKIN: Yeah, I think instead of epsilon, 00:44:38.090 --> 00:44:40.283 align:middle line:90% it's epsilon over 2. 00:44:40.283 --> 00:44:42.450 align:middle line:84% AUDIENCE: So but then if the epsilons are shrinking, 00:44:42.450 --> 00:44:44.283 align:middle line:84% how are you getting arbitrarily [INAUDIBLE]? 00:44:44.283 --> 00:44:46.158 align:middle line:84% PABLO SHMERKIN: Yeah, that's a good question. 00:44:46.158 --> 00:44:46.753 align:middle line:90% So OK. 00:44:46.753 --> 00:44:48.880 align:middle line:84% Epsilon, in some sense, is shrinking. 00:44:48.880 --> 00:44:52.060 align:middle line:84% But OK, so there is one epsilon, which 00:44:52.060 --> 00:44:55.500 align:middle line:84% is the epsilon for the size of A plus little aA. 00:44:55.500 --> 00:44:58.460 align:middle line:84% And there is potentially a different epsilon 00:44:58.460 --> 00:45:01.060 align:middle line:84% if we want to find a delta S plus epsilon 00:45:01.060 --> 00:45:03.620 align:middle line:90% set inside a plus little aA. 00:45:03.620 --> 00:45:05.680 align:middle line:84% Those two epsilons, I think one is epsilon, 00:45:05.680 --> 00:45:07.700 align:middle line:90% the other is epsilon over 2. 00:45:07.700 --> 00:45:10.400 align:middle line:84% But the point is that they depend on S continuously, 00:45:10.400 --> 00:45:12.540 align:middle line:90% both of them. 00:45:12.540 --> 00:45:15.980 align:middle line:84% So they can be taken to be continuous functions of S. 00:45:15.980 --> 00:45:18.100 align:middle line:84% Because they are continuous functions of S 00:45:18.100 --> 00:45:21.900 align:middle line:84% as long as S is less than 1 minus tau, 00:45:21.900 --> 00:45:24.500 align:middle line:90% S is bounded away from 0. 00:45:24.500 --> 00:45:27.390 align:middle line:84% So we reach 1 minus tau in finitely many steps. 00:45:27.390 --> 00:45:30.540 align:middle line:90% 00:45:30.540 --> 00:45:32.960 align:middle line:84% Maybe let's write this here because it's important. 00:45:32.960 --> 00:45:46.740 align:middle line:84% So epsilon is, or can be taken, continuous function of S. 00:45:46.740 --> 00:46:01.400 align:middle line:84% So remains bounded below as long as S is bounded away from 1. 00:46:01.400 --> 00:46:07.100 align:middle line:90% 00:46:07.100 --> 00:46:10.480 align:middle line:84% So we can achieve this by iterating finitely many times. 00:46:10.480 --> 00:46:16.620 align:middle line:90% 00:46:16.620 --> 00:46:18.520 align:middle line:90% OK, and why is this good? 00:46:18.520 --> 00:46:22.620 align:middle line:84% Because this allows us to do instead of A plus aA, 00:46:22.620 --> 00:46:27.140 align:middle line:84% now we can do x plus aX, where x is potentially much 00:46:27.140 --> 00:46:28.240 align:middle line:90% bigger than a. 00:46:28.240 --> 00:46:33.540 align:middle line:90% 00:46:33.540 --> 00:46:38.220 align:middle line:84% So the next step, and I think this step is the same as 00:46:38.220 --> 00:46:39.400 align:middle line:90% in the finite-field setting. 00:46:39.400 --> 00:46:42.110 align:middle line:84% So I'm basically just going to state it. 00:46:42.110 --> 00:46:46.870 align:middle line:90% If x is a delta t-- 00:46:46.870 --> 00:46:48.780 align:middle line:90% OK, x is 0, 1 still. 00:46:48.780 --> 00:46:53.150 align:middle line:90% 00:46:53.150 --> 00:46:57.950 align:middle line:90% And x is a delta. 00:46:57.950 --> 00:46:59.070 align:middle line:90% Maybe let's call it u. 00:46:59.070 --> 00:47:02.110 align:middle line:84% So u is going to be t over tau soon. 00:47:02.110 --> 00:47:13.910 align:middle line:84% Delta t to the minus epsilon set on A is 0,1. 00:47:13.910 --> 00:47:18.830 align:middle line:84% It's delta S. Maybe let's write eta here for consistency. 00:47:18.830 --> 00:47:21.230 align:middle line:90% Delta to the minus eta set. 00:47:21.230 --> 00:47:24.070 align:middle line:84% So now, instead of one set, we have two sets potentially 00:47:24.070 --> 00:47:26.070 align:middle line:90% of very different sizes. 00:47:26.070 --> 00:47:30.510 align:middle line:84% And OK, they are both strictly bigger than 0, 00:47:30.510 --> 00:47:32.310 align:middle line:90% and strictly less than 1. 00:47:32.310 --> 00:47:38.670 align:middle line:84% And the interesting case is when S is less than u. 00:47:38.670 --> 00:47:41.260 align:middle line:84% So this is the case that we don't already know. 00:47:41.260 --> 00:47:46.840 align:middle line:90% 00:47:46.840 --> 00:47:53.930 align:middle line:84% Then there exists little A in A such that x plus little aX 00:47:53.930 --> 00:47:54.430 align:middle line:90% grows. 00:47:54.430 --> 00:48:00.800 align:middle line:90% 00:48:00.800 --> 00:48:04.620 align:middle line:84% It grows by some epsilon that depends on S and u. 00:48:04.620 --> 00:48:16.051 align:middle line:90% 00:48:16.051 --> 00:48:20.860 align:middle line:84% OK, so just briefly, so the idea, 00:48:20.860 --> 00:48:22.333 align:middle line:84% and again, I think you've seen this 00:48:22.333 --> 00:48:23.500 align:middle line:90% in the finite-field setting. 00:48:23.500 --> 00:48:31.080 align:middle line:84% But the idea is that it's true for maybe not for A, 00:48:31.080 --> 00:48:40.930 align:middle line:84% but it's true for Q tau of A. And here, you 00:48:40.930 --> 00:48:47.210 align:middle line:84% have to take tau so that Q tau of a is bigger than x. 00:48:47.210 --> 00:48:52.450 align:middle line:84% So we can take tau, for example, to be 1 minus u over 2. 00:48:52.450 --> 00:48:53.850 align:middle line:90% Something like this. 00:48:53.850 --> 00:48:58.300 align:middle line:84% Then this is much bigger than x in size. 00:48:58.300 --> 00:48:59.050 align:middle line:90% So these are sets. 00:48:59.050 --> 00:49:02.330 align:middle line:84% But it has size much bigger than x. 00:49:02.330 --> 00:49:07.070 align:middle line:84% Once the set of directions has size much bigger than x, 00:49:07.070 --> 00:49:10.650 align:middle line:84% we can do double counting and get growth 00:49:10.650 --> 00:49:12.330 align:middle line:90% with this instead of this. 00:49:12.330 --> 00:49:14.570 align:middle line:84% But once we get growth for a polynomial instead of A, 00:49:14.570 --> 00:49:17.050 align:middle line:84% we apply Plunnecke-Ruzsa one million times, 00:49:17.050 --> 00:49:30.010 align:middle line:84% and we go back down to A. OK, then apply 00:49:30.010 --> 00:49:36.370 align:middle line:84% Plunnecke-Ruzsa triangle inequality, double counting. 00:49:36.370 --> 00:49:44.140 align:middle line:90% 00:49:44.140 --> 00:49:49.820 align:middle line:90% And then it is also true for A. 00:49:49.820 --> 00:49:52.600 align:middle line:84% So this part is really exactly the same. 00:49:52.600 --> 00:49:58.700 align:middle line:84% So we already had to go from Q of A to little a before. 00:49:58.700 --> 00:50:02.340 align:middle line:84% So initially, we proved that aA A minus aA A is bigger than A. 00:50:02.340 --> 00:50:05.700 align:middle line:84% And we use that to show that A plus little aA is bigger than A. 00:50:05.700 --> 00:50:08.463 align:middle line:90% So it's the same reduction here. 00:50:08.463 --> 00:50:10.380 align:middle line:84% And I think you've seen something very similar 00:50:10.380 --> 00:50:14.100 align:middle line:84% in the finite-field setting and modulo technical details. 00:50:14.100 --> 00:50:17.740 align:middle line:90% This part is the same. 00:50:17.740 --> 00:50:20.300 align:middle line:84% But in order to get here, it's important 00:50:20.300 --> 00:50:23.420 align:middle line:84% that we can make this set bigger than x. 00:50:23.420 --> 00:50:25.120 align:middle line:84% This is what we wanted to iterate. 00:50:25.120 --> 00:50:27.660 align:middle line:90% 00:50:27.660 --> 00:50:29.640 align:middle line:84% Potentially, x is much bigger than A, 00:50:29.640 --> 00:50:32.800 align:middle line:84% but we can expand A by a polynomial 00:50:32.800 --> 00:50:34.950 align:middle line:84% so that it becomes bigger than x. 00:50:34.950 --> 00:50:36.530 align:middle line:90% And for this, we had to iterate. 00:50:36.530 --> 00:50:39.810 align:middle line:84% This is why we worried so much about iterating. 00:50:39.810 --> 00:50:42.310 align:middle line:90% 00:50:42.310 --> 00:50:42.810 align:middle line:90% OK. 00:50:42.810 --> 00:50:47.150 align:middle line:90% 00:50:47.150 --> 00:50:52.760 align:middle line:84% OK, the next step is to prove Bourgain's prediction theorem. 00:50:52.760 --> 00:50:56.470 align:middle line:90% 00:50:56.470 --> 00:50:58.450 align:middle line:84% OK, I'm not going to do the full proof, 00:50:58.450 --> 00:50:59.870 align:middle line:84% but just basically recall what you 00:50:59.870 --> 00:51:04.310 align:middle line:84% did in the finite-field setting, and explain, again, where one 00:51:04.310 --> 00:51:06.190 align:middle line:90% has to be a little bit careful. 00:51:06.190 --> 00:51:10.030 align:middle line:84% But hopefully, by now, you will not be so worried about the part 00:51:10.030 --> 00:51:11.810 align:middle line:90% where you have to be careful. 00:51:11.810 --> 00:51:13.560 align:middle line:84% So proof of Bourgain's projection theorem. 00:51:13.560 --> 00:51:17.790 align:middle line:90% 00:51:17.790 --> 00:51:20.630 align:middle line:84% OK, now I regret calling this xx. 00:51:20.630 --> 00:51:24.030 align:middle line:90% But it's too late. 00:51:24.030 --> 00:51:33.356 align:middle line:84% So let x be in the unit ball of R2 delta t delta 00:51:33.356 --> 00:51:37.320 align:middle line:84% to the minus eta set, where eta is very, very small. 00:51:37.320 --> 00:51:39.220 align:middle line:90% And t is between 0 and 2. 00:51:39.220 --> 00:51:43.720 align:middle line:90% 00:51:43.720 --> 00:51:54.080 align:middle line:84% Let D be a set of directions, which is a delta S 00:51:54.080 --> 00:51:55.940 align:middle line:90% delta to the minus eta set. 00:51:55.940 --> 00:52:03.240 align:middle line:90% 00:52:03.240 --> 00:52:04.330 align:middle line:90% And what is the goal? 00:52:04.330 --> 00:52:06.973 align:middle line:90% 00:52:06.973 --> 00:52:09.140 align:middle line:84% So what is the claim of Bourgain projection theorem? 00:52:09.140 --> 00:52:20.080 align:middle line:84% It is that there exists some a and D such that pi a of G 00:52:20.080 --> 00:52:24.070 align:middle line:84% exceeds the trivial bound by some epsilon. 00:52:24.070 --> 00:52:28.120 align:middle line:90% 00:52:28.120 --> 00:52:33.270 align:middle line:84% And the trivial bound is the square root of the delta. 00:52:33.270 --> 00:52:34.250 align:middle line:90% OK. 00:52:34.250 --> 00:52:34.750 align:middle line:90% Sorry. 00:52:34.750 --> 00:52:38.290 align:middle line:84% In the version that I stated, the size 00:52:38.290 --> 00:52:41.630 align:middle line:90% of x is delta to the minus t. 00:52:41.630 --> 00:52:44.190 align:middle line:84% So it matches the non-concentration condition. 00:52:44.190 --> 00:52:47.210 align:middle line:90% 00:52:47.210 --> 00:52:50.410 align:middle line:84% So here, we can just write delta to the minus t over 2. 00:52:50.410 --> 00:52:52.510 align:middle line:84% This is the square root of the size of x, 00:52:52.510 --> 00:52:53.790 align:middle line:90% which is the trivial bound. 00:52:53.790 --> 00:52:56.890 align:middle line:84% And we exceed it by some delta to the minus epsilon. 00:52:56.890 --> 00:53:07.850 align:middle line:84% And this is true for every G, which is dense in A times A. 00:53:07.850 --> 00:53:10.150 align:middle line:84% OK, so this is Bourgain's projection theorem. 00:53:10.150 --> 00:53:12.096 align:middle line:90% This is what we want to prove. 00:53:12.096 --> 00:53:15.010 align:middle line:84% OK, so in the finite-field setting, 00:53:15.010 --> 00:53:18.210 align:middle line:84% I think to prove a similar statement, 00:53:18.210 --> 00:53:21.610 align:middle line:84% what you did is first consider three fix-- 00:53:21.610 --> 00:53:26.290 align:middle line:84% well, just pick three directions in D. 00:53:26.290 --> 00:53:28.470 align:middle line:84% In order to apply Balog-Szemeredi-Gowers. 00:53:28.470 --> 00:53:30.678 align:middle line:84% So again, it's similar to what we did before. 00:53:30.678 --> 00:53:32.220 align:middle line:84% Either things already work, or we can 00:53:32.220 --> 00:53:33.885 align:middle line:90% apply Balog-Szemeredi-Gowers. 00:53:33.885 --> 00:53:36.260 align:middle line:84% Here, we have to be careful about how we choose the three 00:53:36.260 --> 00:53:39.820 align:middle line:84% directions, because if the three directions that we choose 00:53:39.820 --> 00:53:42.960 align:middle line:84% are very close to each other, this is going to be bad news. 00:53:42.960 --> 00:53:45.780 align:middle line:90% We're going to lose a lot. 00:53:45.780 --> 00:53:49.380 align:middle line:84% So well, even though this is 0,1, 00:53:49.380 --> 00:54:02.460 align:middle line:84% we can sort of change coordinates so that 0,1 00:54:02.460 --> 00:54:06.460 align:middle line:90% and infinity are in D. 00:54:06.460 --> 00:54:09.460 align:middle line:84% And this can be done in a way where 00:54:09.460 --> 00:54:15.020 align:middle line:84% we don't lose too much using that D is spread out. 00:54:15.020 --> 00:54:18.140 align:middle line:84% Because of this non-concentration conditions, 00:54:18.140 --> 00:54:19.540 align:middle line:84% there are three points in D which 00:54:19.540 --> 00:54:21.650 align:middle line:90% are far apart from each other. 00:54:21.650 --> 00:54:23.900 align:middle line:84% And that means that if we send these three points that 00:54:23.900 --> 00:54:25.660 align:middle line:84% are far apart from each other to 0, 1, 00:54:25.660 --> 00:54:28.490 align:middle line:84% infinity, so these are really the slopes. 00:54:28.490 --> 00:54:29.490 align:middle line:90% They are not the angles. 00:54:29.490 --> 00:54:31.410 align:middle line:90% They are slopes. 00:54:31.410 --> 00:54:47.990 align:middle line:84% So this can be done with a linear change of coordinates 00:54:47.990 --> 00:54:49.250 align:middle line:90% of distortion. 00:54:49.250 --> 00:54:51.910 align:middle line:90% 00:54:51.910 --> 00:54:56.110 align:middle line:84% So the distortion is the norm of the corresponding matrix 00:54:56.110 --> 00:55:00.350 align:middle line:90% delta to the minus O of eta. 00:55:00.350 --> 00:55:02.310 align:middle line:84% So this distortion is, because eta 00:55:02.310 --> 00:55:04.870 align:middle line:90% is much smaller than epsilon. 00:55:04.870 --> 00:55:08.813 align:middle line:84% If we have a gain, well, we have to decrease the gain 00:55:08.813 --> 00:55:10.730 align:middle line:84% by what we lose in this change of coordinates. 00:55:10.730 --> 00:55:11.730 align:middle line:90% But this is mild. 00:55:11.730 --> 00:55:24.335 align:middle line:90% 00:55:24.335 --> 00:55:29.220 align:middle line:84% AUDIENCE: [INAUDIBLE] A to be pi, 0, union pi infinity of x? 00:55:29.220 --> 00:55:31.372 align:middle line:90% 00:55:31.372 --> 00:55:32.330 align:middle line:90% PABLO SHMERKIN: We are. 00:55:32.330 --> 00:55:34.630 align:middle line:84% We are going to look at the vertical and horizontal 00:55:34.630 --> 00:55:38.525 align:middle line:84% projections of x and the pi 1 of x. 00:55:38.525 --> 00:55:40.150 align:middle line:84% AUDIENCE: Yeah, I'm just saying there's 00:55:40.150 --> 00:55:42.917 align:middle line:84% A cross A in the statement of the theorem. 00:55:42.917 --> 00:55:44.000 align:middle line:90% PABLO SHMERKIN: Oh, sorry. 00:55:44.000 --> 00:55:48.210 align:middle line:90% 00:55:48.210 --> 00:55:48.710 align:middle line:90% Sorry. 00:55:48.710 --> 00:55:49.210 align:middle line:90% In x. 00:55:49.210 --> 00:55:50.710 align:middle line:90% In x. 00:55:50.710 --> 00:55:51.550 align:middle line:90% Sorry. 00:55:51.550 --> 00:55:54.230 align:middle line:90% AUDIENCE: X cross x? 00:55:54.230 --> 00:55:55.180 align:middle line:90% PABLO SHMERKIN: In x. 00:55:55.180 --> 00:55:55.940 align:middle line:90% x is in R2. 00:55:55.940 --> 00:55:59.030 align:middle line:90% 00:55:59.030 --> 00:56:00.710 align:middle line:90% In x. 00:56:00.710 --> 00:56:02.320 align:middle line:90% x is a two-dimensional object. 00:56:02.320 --> 00:56:03.778 align:middle line:84% AUDIENCE: And the point is you want 00:56:03.778 --> 00:56:05.530 align:middle line:84% to try to prove this using A equals sub pi 00:56:05.530 --> 00:56:08.590 align:middle line:90% 0, union pi infinity x? 00:56:08.590 --> 00:56:09.710 align:middle line:90% PABLO SHMERKIN: Yes. 00:56:09.710 --> 00:56:10.670 align:middle line:90% Yeah, that's the point. 00:56:10.670 --> 00:56:11.170 align:middle line:90% But yeah. 00:56:11.170 --> 00:56:11.670 align:middle line:90% So sorry. 00:56:11.670 --> 00:56:13.270 align:middle line:90% There was a typo here. 00:56:13.270 --> 00:56:15.670 align:middle line:90% So x is already two dimensional. 00:56:15.670 --> 00:56:18.390 align:middle line:84% And what we have been doing, we have a Cartesian product. 00:56:18.390 --> 00:56:20.830 align:middle line:90% Or here, we have a Cartesian-- 00:56:20.830 --> 00:56:21.510 align:middle line:90% OK. 00:56:21.510 --> 00:56:24.920 align:middle line:90% But [INAUDIBLE]. 00:56:24.920 --> 00:56:28.220 align:middle line:84% So OK, we want to apply this, which unfortunately, 00:56:28.220 --> 00:56:29.520 align:middle line:90% is going to be hidden. 00:56:29.520 --> 00:56:33.520 align:middle line:84% OK, so x, this x, is going to be some-- the projection of x, 00:56:33.520 --> 00:56:35.430 align:middle line:84% horizontal projection of the x in R2. 00:56:35.430 --> 00:56:39.215 align:middle line:90% 00:56:39.215 --> 00:56:41.340 align:middle line:84% So it's really similar to the finite-field setting. 00:56:41.340 --> 00:56:43.520 align:middle line:90% So I'm skipping details. 00:56:43.520 --> 00:56:45.400 align:middle line:84% I'm going to continue skipping details, 00:56:45.400 --> 00:56:47.145 align:middle line:84% mostly focusing on the differences. 00:56:47.145 --> 00:56:48.520 align:middle line:84% So one difference is that we have 00:56:48.520 --> 00:56:51.720 align:middle line:84% to be careful with the three directions that we take. 00:56:51.720 --> 00:56:53.625 align:middle line:84% If they are too close to each other, 00:56:53.625 --> 00:56:55.000 align:middle line:84% the distortion will be too large, 00:56:55.000 --> 00:56:58.820 align:middle line:84% and it can be so large that we lose all the gain. 00:56:58.820 --> 00:57:01.360 align:middle line:84% But because the set of directions 00:57:01.360 --> 00:57:03.460 align:middle line:84% is spread out by the non-concentration condition, 00:57:03.460 --> 00:57:06.140 align:middle line:84% we can do it in such a way that the distortion is controlled. 00:57:06.140 --> 00:57:20.840 align:middle line:90% 00:57:20.840 --> 00:57:25.610 align:middle line:84% OK, so now, x is contained in x1 times x2, where 00:57:25.610 --> 00:57:28.890 align:middle line:84% this is the horizontal projection, 00:57:28.890 --> 00:57:30.710 align:middle line:84% and this is the vertical projection. 00:57:30.710 --> 00:57:35.010 align:middle line:90% 00:57:35.010 --> 00:57:44.290 align:middle line:84% So if, because we are assuming that also one is in D, if one 00:57:44.290 --> 00:57:47.010 align:middle line:90% works, we are done. 00:57:47.010 --> 00:57:49.930 align:middle line:84% Because we are trying to show that something works. 00:57:49.930 --> 00:57:52.170 align:middle line:84% So the claim is that there exists an A and D 00:57:52.170 --> 00:57:54.130 align:middle line:90% with this property. 00:57:54.130 --> 00:57:57.510 align:middle line:84% So if we are so lucky that one has this property, we are done. 00:57:57.510 --> 00:58:03.130 align:middle line:90% 00:58:03.130 --> 00:58:07.050 align:middle line:84% If one doesn't work, so this is very 00:58:07.050 --> 00:58:09.970 align:middle line:84% similar to what we did a bit earlier, but somehow, 00:58:09.970 --> 00:58:11.430 align:middle line:90% we have to do it twice. 00:58:11.430 --> 00:58:13.670 align:middle line:84% Once to do the iteration for A plus little aA, 00:58:13.670 --> 00:58:16.490 align:middle line:84% and then again, to conclude the proof. 00:58:16.490 --> 00:58:21.020 align:middle line:84% If one doesn't work, then there exists some G, 00:58:21.020 --> 00:58:22.890 align:middle line:90% which is dense in x1 times x2. 00:58:22.890 --> 00:58:25.540 align:middle line:90% 00:58:25.540 --> 00:58:26.760 align:middle line:90% OK, sorry. 00:58:26.760 --> 00:58:29.078 align:middle line:90% Why is it dense in x1? 00:58:29.078 --> 00:58:29.578 align:middle line:90% Ah. 00:58:29.578 --> 00:58:32.980 align:middle line:90% 00:58:32.980 --> 00:58:36.268 align:middle line:90% OK, actually, yeah. 00:58:36.268 --> 00:58:37.310 align:middle line:90% This is more complicated. 00:58:37.310 --> 00:58:39.320 align:middle line:90% So one has to be careful. 00:58:39.320 --> 00:58:39.920 align:middle line:90% So OK. 00:58:39.920 --> 00:58:49.720 align:middle line:90% So OK. 00:58:49.720 --> 00:58:54.380 align:middle line:84% So I want to assume that x1 has size 00:58:54.380 --> 00:58:57.980 align:middle line:84% roughly square root of the size of x with the idea 00:58:57.980 --> 00:59:02.660 align:middle line:84% that if it had size bigger than the size of x, we would win. 00:59:02.660 --> 00:59:05.295 align:middle line:84% But in fact, in order to reach this conclusion, 00:59:05.295 --> 00:59:07.420 align:middle line:84% we have to apply the electromagnetic hours already, 00:59:07.420 --> 00:59:10.260 align:middle line:84% because the conclusion we want is not 00:59:10.260 --> 00:59:15.160 align:middle line:84% only that the projection of x grows, 00:59:15.160 --> 00:59:18.260 align:middle line:84% but the projection of every dense subset of x works. 00:59:18.260 --> 00:59:21.630 align:middle line:90% So OK. 00:59:21.630 --> 00:59:24.910 align:middle line:84% So one really has to do is apply value of electromagnetic hours 00:59:24.910 --> 00:59:25.990 align:middle line:90% three times. 00:59:25.990 --> 00:59:30.270 align:middle line:84% So maybe for 0, infinity, and 1, in this order, 00:59:30.270 --> 00:59:36.950 align:middle line:84% either it works or we apply Balog-Szemeredi-Gowers 00:59:36.950 --> 00:59:39.710 align:middle line:84% to replace x by x prime, and then x double prime, 00:59:39.710 --> 00:59:41.790 align:middle line:90% and x triple prime. 00:59:41.790 --> 00:59:44.750 align:middle line:84% OK, I'm not going to do all of the details, 00:59:44.750 --> 00:59:49.803 align:middle line:84% but did something like this happen 00:59:49.803 --> 00:59:50.970 align:middle line:90% in the finite-field setting? 00:59:50.970 --> 00:59:53.150 align:middle line:84% I think it also, this part has to be done 00:59:53.150 --> 00:59:56.150 align:middle line:90% in the finite-field setting. 00:59:56.150 --> 00:59:57.830 align:middle line:90% So-- 00:59:57.830 --> 01:00:01.192 align:middle line:84% AUDIENCE: We didn't do this strong version in detail. 01:00:01.192 --> 01:00:02.150 align:middle line:90% PABLO SHMERKIN: Oh, OK. 01:00:02.150 --> 01:00:03.900 align:middle line:84% We didn't do the strong version in detail. 01:00:03.900 --> 01:00:07.190 align:middle line:84% OK, also, I'm not going to do the strong version in detail 01:00:07.190 --> 01:00:09.550 align:middle line:84% because the 20 minutes I expected to take 01:00:09.550 --> 01:00:11.790 align:middle line:84% are turning into the whole lecture again. 01:00:11.790 --> 01:00:13.650 align:middle line:90% But I'm leaving today. 01:00:13.650 --> 01:00:16.630 align:middle line:84% So now, I really have to finish it over the next 20 minutes. 01:00:16.630 --> 01:00:18.720 align:middle line:90% So OK. 01:00:18.720 --> 01:00:27.320 align:middle line:84% So OK, either 0, 1, or infinity, or I 01:00:27.320 --> 01:00:40.600 align:middle line:84% guess 0, infinity, or 1 work, or in each case, 01:00:40.600 --> 01:00:49.560 align:middle line:84% we can apply Balog-Szemeredi-Gowers and pass 01:00:49.560 --> 01:01:02.160 align:middle line:84% to dense subsets of x of x and the corresponding projections 01:01:02.160 --> 01:01:02.780 align:middle line:90% of x. 01:01:02.780 --> 01:01:07.720 align:middle line:90% 01:01:07.720 --> 01:01:09.460 align:middle line:90% So Balog-Szemeredi-Gowers-- OK. 01:01:09.460 --> 01:01:11.980 align:middle line:90% So let's take 0. 01:01:11.980 --> 01:01:15.850 align:middle line:84% If the horizontal projection has the property we want, 01:01:15.850 --> 01:01:17.030 align:middle line:90% we are done. 01:01:17.030 --> 01:01:19.330 align:middle line:84% Otherwise, there exists a dense subset of A 01:01:19.330 --> 01:01:21.007 align:middle line:84% with a small horizontal projection. 01:01:21.007 --> 01:01:23.090 align:middle line:84% But you can imagine that the horizontal projection 01:01:23.090 --> 01:01:24.470 align:middle line:90% is another projection. 01:01:24.470 --> 01:01:29.090 align:middle line:84% So it's the same thing we did last time, we did a bit earlier. 01:01:29.090 --> 01:01:32.650 align:middle line:84% And then we have to replace x by some x prime, and then by some 01:01:32.650 --> 01:01:35.490 align:middle line:84% x double prime, and then by some x triple prime. 01:01:35.490 --> 01:01:40.530 align:middle line:84% So I'm not going to do it in detail. 01:01:40.530 --> 01:01:41.690 align:middle line:90% I'm sorry. 01:01:41.690 --> 01:01:43.790 align:middle line:84% Yeah, I see many of you are confused. 01:01:43.790 --> 01:01:47.130 align:middle line:84% And that's OK because one has to do it carefully. 01:01:47.130 --> 01:01:49.970 align:middle line:84% It's not obvious how to do it, but it can be done. 01:01:49.970 --> 01:02:03.930 align:middle line:84% OK, so eventually, after renaming the set back to x, 01:02:03.930 --> 01:02:15.660 align:middle line:84% we are in the following situation, this situation where, 01:02:15.660 --> 01:02:21.340 align:middle line:84% OK, let me just write it here, where 01:02:21.340 --> 01:02:28.700 align:middle line:84% the horizontal and vertical projections have size, at most, 01:02:28.700 --> 01:02:33.270 align:middle line:84% delta to the minus epsilon, t over 2 plus epsilon. 01:02:33.270 --> 01:02:46.500 align:middle line:90% 01:02:46.500 --> 01:02:52.740 align:middle line:84% OK, I guess let's say we apply Balog-Szemeredi-Gowers twice. 01:02:52.740 --> 01:02:55.070 align:middle line:84% So we don't apply it to pi 1 yet. 01:02:55.070 --> 01:03:04.700 align:middle line:90% 01:03:04.700 --> 01:03:07.460 align:middle line:84% OK, just to see how this goes, so suppose 01:03:07.460 --> 01:03:12.910 align:middle line:84% that pi 1, so suppose that one doesn't work. 01:03:12.910 --> 01:03:14.710 align:middle line:90% Suppose that one doesn't work. 01:03:14.710 --> 01:03:21.630 align:middle line:84% And that means that pi 1 of G doesn't grow 01:03:21.630 --> 01:03:25.470 align:middle line:90% for some dense subset of x. 01:03:25.470 --> 01:03:27.230 align:middle line:84% And if it's a dense subset of x, it 01:03:27.230 --> 01:03:29.850 align:middle line:84% is a dense subset of x1 times x2. 01:03:29.850 --> 01:03:30.530 align:middle line:90% This is x1. 01:03:30.530 --> 01:03:31.040 align:middle line:90% This is x2. 01:03:31.040 --> 01:03:49.430 align:middle line:90% 01:03:49.430 --> 01:03:50.490 align:middle line:90% So this is x1. 01:03:50.490 --> 01:03:51.000 align:middle line:90% This is x2. 01:03:51.000 --> 01:03:56.950 align:middle line:90% 01:03:56.950 --> 01:03:59.850 align:middle line:84% OK, then we're going to apply Balog-Szemeredi-Gowers. 01:03:59.850 --> 01:04:01.710 align:middle line:84% So here, you see that we are in this setting 01:04:01.710 --> 01:04:02.930 align:middle line:90% to apply Balog-Szemeredi-Gowers. 01:04:02.930 --> 01:04:04.888 align:middle line:84% So we're going to apply Balog-Szemeredi-Gowers. 01:04:04.888 --> 01:04:09.570 align:middle line:84% So it looks like after applying Balog-Szemeredi-Gowers 01:04:09.570 --> 01:04:12.180 align:middle line:84% lots of times, and I skipped the details, 01:04:12.180 --> 01:04:14.400 align:middle line:84% but you have to trust me that this can be done, 01:04:14.400 --> 01:04:19.280 align:middle line:84% we are in the setting where we can apply this and be done. 01:04:19.280 --> 01:04:22.240 align:middle line:90% But not so fast. 01:04:22.240 --> 01:04:23.480 align:middle line:90% What's the issue? 01:04:23.480 --> 01:04:24.410 align:middle line:90% Why not so fast? 01:04:24.410 --> 01:04:27.240 align:middle line:90% 01:04:27.240 --> 01:04:29.760 align:middle line:84% So all of the part that I'm not explaining here, 01:04:29.760 --> 01:04:32.060 align:middle line:84% you also have to do it in the finite-field setting 01:04:32.060 --> 01:04:35.880 align:middle line:84% if you want to get the stronger statement that the projection 01:04:35.880 --> 01:04:39.890 align:middle line:84% of a dense subset grows, not a dense subset of the given set. 01:04:39.890 --> 01:04:42.140 align:middle line:84% If you want to prove that in the finite-field setting, 01:04:42.140 --> 01:04:44.660 align:middle line:84% you have to do this multiple Balog-Szemeredi-Gowers. 01:04:44.660 --> 01:04:46.160 align:middle line:84% So this is not a difference, OK, you 01:04:46.160 --> 01:04:48.410 align:middle line:84% haven't done it in detail in the finite-field setting. 01:04:48.410 --> 01:04:50.680 align:middle line:84% But this part is not a difference 01:04:50.680 --> 01:04:53.960 align:middle line:84% between the finite-field setting and the Euclidean setting. 01:04:53.960 --> 01:04:56.380 align:middle line:84% But there is something which is a difference. 01:04:56.380 --> 01:04:57.960 align:middle line:90% So what is the issue? 01:04:57.960 --> 01:04:59.820 align:middle line:90% Why can't we just apply, oh, OK. 01:04:59.820 --> 01:05:01.100 align:middle line:90% We can apply this. 01:05:01.100 --> 01:05:03.700 align:middle line:84% But why do we have to be careful about applying this? 01:05:03.700 --> 01:05:13.440 align:middle line:90% 01:05:13.440 --> 01:05:14.946 align:middle line:90% Yes? 01:05:14.946 --> 01:05:17.320 align:middle line:84% AUDIENCE: So if we apply Balog-Szemeredi-Gowers, 01:05:17.320 --> 01:05:20.320 align:middle line:90% we get A prime plus A prime. 01:05:20.320 --> 01:05:25.220 align:middle line:90% And then [INAUDIBLE] y cross y. 01:05:25.220 --> 01:05:27.560 align:middle line:84% That covers a lot of G. And then if I 01:05:27.560 --> 01:05:29.500 align:middle line:84% want to apply what's on the left board, 01:05:29.500 --> 01:05:32.920 align:middle line:84% I would need to know that y is a delta u something set. 01:05:32.920 --> 01:05:33.920 align:middle line:90% PABLO SHMERKIN: Exactly. 01:05:33.920 --> 01:05:37.340 align:middle line:84% So here, we have a Cartesian product. 01:05:37.340 --> 01:05:40.080 align:middle line:84% OK, so one small issue is that instead of x1 times x1, 01:05:40.080 --> 01:05:41.600 align:middle line:90% we have x1 times x2. 01:05:41.600 --> 01:05:43.200 align:middle line:90% But this is not important. 01:05:43.200 --> 01:05:47.692 align:middle line:84% The issue is that x1 is a projection of x. 01:05:47.692 --> 01:05:51.320 align:middle line:84% And we are assuming that the size is roughly the square root 01:05:51.320 --> 01:05:53.280 align:middle line:90% of the size of x. 01:05:53.280 --> 01:05:55.140 align:middle line:90% But why is it not concentrated? 01:05:55.140 --> 01:05:57.200 align:middle line:84% In order to apply this and conclude, 01:05:57.200 --> 01:05:59.760 align:middle line:84% we need to know that x is not concentrated 01:05:59.760 --> 01:06:01.660 align:middle line:90% with u equals to t over 2. 01:06:01.660 --> 01:06:05.000 align:middle line:90% 01:06:05.000 --> 01:06:06.760 align:middle line:84% So it's very similar to the issue 01:06:06.760 --> 01:06:10.490 align:middle line:84% that we had when we needed to iterate. 01:06:10.490 --> 01:06:13.170 align:middle line:84% So we knew that a plus little aA is large. 01:06:13.170 --> 01:06:16.210 align:middle line:84% But we didn't know that the non-concentration condition also 01:06:16.210 --> 01:06:17.330 align:middle line:90% improves. 01:06:17.330 --> 01:06:19.410 align:middle line:90% It's a similar issue here. 01:06:19.410 --> 01:06:22.290 align:middle line:84% We know something about the size of these projections, 01:06:22.290 --> 01:06:23.890 align:middle line:84% but a priori, we don't know anything 01:06:23.890 --> 01:06:26.930 align:middle line:90% about the non-concentrated. 01:06:26.930 --> 01:06:29.730 align:middle line:84% We need these projections to be non-concentrated. 01:06:29.730 --> 01:06:35.050 align:middle line:84% OK, so maybe ignore all of this, and just say the following. 01:06:35.050 --> 01:06:38.890 align:middle line:84% Using Balog-Szemeredi-Gowers many times in a clever way, 01:06:38.890 --> 01:06:43.803 align:middle line:84% we can reduce to the case where the set x is a product set. 01:06:43.803 --> 01:06:45.970 align:middle line:84% You can even assume it's a product set of something, 01:06:45.970 --> 01:06:47.530 align:middle line:90% a self product set. 01:06:47.530 --> 01:06:49.810 align:middle line:84% So using Balog-Szemeredi-Gowers as many times, 01:06:49.810 --> 01:06:53.570 align:middle line:84% we can assume that x is x1 times x2. 01:06:53.570 --> 01:06:58.152 align:middle line:84% And then using Balog-Szemeredi-Gowers again, 01:06:58.152 --> 01:06:59.610 align:middle line:84% so Balog-Szemeredi-Gowers allows us 01:06:59.610 --> 01:07:02.250 align:middle line:84% to go back and forth between dense subsets and just 01:07:02.250 --> 01:07:03.370 align:middle line:90% everything. 01:07:03.370 --> 01:07:05.740 align:middle line:90% So this is what allows us to do. 01:07:05.740 --> 01:07:11.220 align:middle line:84% So we can assume that we are in the setting of a product set. 01:07:11.220 --> 01:07:13.700 align:middle line:90% And then we want to apply this. 01:07:13.700 --> 01:07:15.820 align:middle line:84% But not so fast, because we don't 01:07:15.820 --> 01:07:17.620 align:middle line:84% know that the projections satisfy 01:07:17.620 --> 01:07:19.940 align:middle line:84% the non-concentration assumption a priori. 01:07:19.940 --> 01:07:21.740 align:middle line:90% So we have this problem again. 01:07:21.740 --> 01:07:24.100 align:middle line:84% So it's very similar to the problem we had before. 01:07:24.100 --> 01:07:26.860 align:middle line:84% And it is solved in a similar way. 01:07:26.860 --> 01:07:28.340 align:middle line:90% But it has to be solved. 01:07:28.340 --> 01:07:33.700 align:middle line:84% So in some sense, so you can think of it in this way. 01:07:33.700 --> 01:07:37.300 align:middle line:84% We say that t over 2 is a trivial between [INAUDIBLE] 01:07:37.300 --> 01:07:39.487 align:middle line:90% bound for projections. 01:07:39.487 --> 01:07:41.820 align:middle line:84% What we have to show is that this trivial bound actually 01:07:41.820 --> 01:07:43.670 align:middle line:84% holds in the non-concentration sense. 01:07:43.670 --> 01:07:48.860 align:middle line:90% 01:07:48.860 --> 01:07:53.460 align:middle line:84% So here, 0, 1, and infinity are just generic projections. 01:07:53.460 --> 01:07:57.700 align:middle line:84% In fact, if we pick three random points in D, 01:07:57.700 --> 01:08:01.540 align:middle line:84% this will work, because three random points will be separated. 01:08:01.540 --> 01:08:04.270 align:middle line:84% Three random points will not be too concentrated because 01:08:04.270 --> 01:08:05.630 align:middle line:90% of this. 01:08:05.630 --> 01:08:08.670 align:middle line:84% There are not too many points in a single ball. 01:08:08.670 --> 01:08:12.670 align:middle line:84% So if you randomly sample three points in D, 01:08:12.670 --> 01:08:14.550 align:middle line:84% the linear map that makes these points 01:08:14.550 --> 01:08:16.630 align:middle line:84% horizontal, and vertical, and diagonal, 01:08:16.630 --> 01:08:19.910 align:middle line:90% will have small distortion. 01:08:19.910 --> 01:08:24.750 align:middle line:84% So we need to know that the random projection of x 01:08:24.750 --> 01:08:27.830 align:middle line:90% is a delta T over 2 S set. 01:08:27.830 --> 01:08:29.750 align:middle line:84% In some sense, the goal is to show 01:08:29.750 --> 01:08:32.410 align:middle line:84% that the random projection is a delta t over 2 01:08:32.410 --> 01:08:35.413 align:middle line:90% plus some gain S set. 01:08:35.413 --> 01:08:36.830 align:middle line:84% But to prove that, we need to show 01:08:36.830 --> 01:08:39.569 align:middle line:84% that it satisfies the same thing, 01:08:39.569 --> 01:08:42.350 align:middle line:84% but without the epsilon, but the non-concentration version 01:08:42.350 --> 01:08:44.109 align:middle line:90% of that. 01:08:44.109 --> 01:08:45.170 align:middle line:90% And this can be done. 01:08:45.170 --> 01:08:46.990 align:middle line:90% It can be done in several ways. 01:08:46.990 --> 01:08:51.149 align:middle line:84% One way is a similar thing to what we have done today. 01:08:51.149 --> 01:08:53.010 align:middle line:90% Well, today and last time. 01:08:53.010 --> 01:08:59.550 align:middle line:84% But basically, go from growth to non-concentration 01:08:59.550 --> 01:09:03.520 align:middle line:84% by uniformizing everything carefully. 01:09:03.520 --> 01:09:06.260 align:middle line:84% And it can also be done by some careful double counting, 01:09:06.260 --> 01:09:09.915 align:middle line:84% similar to, but not exactly the same that you've done before. 01:09:09.915 --> 01:09:12.040 align:middle line:84% So there are several ways of doing it, because it's 01:09:12.040 --> 01:09:13.479 align:middle line:90% like the easy case. 01:09:13.479 --> 01:09:15.600 align:middle line:90% But it has to be done. 01:09:15.600 --> 01:09:18.290 align:middle line:84% OK, so maybe let's write that down. 01:09:18.290 --> 01:09:33.080 align:middle line:90% 01:09:33.080 --> 01:09:35.469 align:middle line:90% OK, so we need a last fact. 01:09:35.469 --> 01:09:44.800 align:middle line:90% 01:09:44.800 --> 01:09:55.120 align:middle line:84% If x and D are as before, as in Bourgain's projection theorem, 01:09:55.120 --> 01:10:00.240 align:middle line:84% then there exists some A and D. And once again, once we know 01:10:00.240 --> 01:10:03.890 align:middle line:84% that there is one A and D, that means nearly all A and D have 01:10:03.890 --> 01:10:09.890 align:middle line:84% this property such that pi of x contains 01:10:09.890 --> 01:10:18.880 align:middle line:84% a delta t over 2 delta 2, maybe O of epsilon, O of eta set. 01:10:18.880 --> 01:10:26.490 align:middle line:90% 01:10:26.490 --> 01:10:28.468 align:middle line:84% Again, t over 2 is like the trivial bound. 01:10:28.468 --> 01:10:30.010 align:middle line:84% But it's not so trivial in this case, 01:10:30.010 --> 01:10:31.810 align:middle line:84% because we are claiming some concentration 01:10:31.810 --> 01:10:33.210 align:middle line:90% for the projection. 01:10:33.210 --> 01:10:38.030 align:middle line:84% But because at the same time, for covering number, 01:10:38.030 --> 01:10:39.930 align:middle line:84% so for the covering number version of this 01:10:39.930 --> 01:10:43.310 align:middle line:84% is sort of obvious, because again, if we have one set, 01:10:43.310 --> 01:10:45.650 align:middle line:84% you look at two projections in more or less orthogonal 01:10:45.650 --> 01:10:47.570 align:middle line:84% directions, and most one of them can 01:10:47.570 --> 01:10:50.490 align:middle line:84% drop by more than square root of the size of the set. 01:10:50.490 --> 01:10:53.410 align:middle line:90% And this is true at every scale. 01:10:53.410 --> 01:10:55.470 align:middle line:84% So you see that if we knew that this is uniform, 01:10:55.470 --> 01:10:56.450 align:middle line:90% we would be done. 01:10:56.450 --> 01:11:00.140 align:middle line:84% So it's really very similar to what we have been discussing 01:11:00.140 --> 01:11:02.500 align:middle line:90% in the rest of the lecture. 01:11:02.500 --> 01:11:18.700 align:middle line:84% So this can be proved using a similar strategy 01:11:18.700 --> 01:11:22.460 align:middle line:90% as for A plus aA. 01:11:22.460 --> 01:11:27.620 align:middle line:84% Or it can also be done without using uniformization 01:11:27.620 --> 01:11:28.920 align:middle line:90% by double counting. 01:11:28.920 --> 01:11:29.980 align:middle line:90% So OK. 01:11:29.980 --> 01:11:33.760 align:middle line:90% 01:11:33.760 --> 01:11:35.597 align:middle line:90% OK, maybe not uniformization. 01:11:35.597 --> 01:11:37.180 align:middle line:84% But if one doesn't use uniformization, 01:11:37.180 --> 01:11:39.740 align:middle line:84% maybe one needs some of Ruzsa's lemma. 01:11:39.740 --> 01:11:42.700 align:middle line:90% But OK, so it can be done. 01:11:42.700 --> 01:11:45.820 align:middle line:84% But I just wanted to point out that this is yet 01:11:45.820 --> 01:11:49.040 align:middle line:84% another place where we need to worry about the fact that, 01:11:49.040 --> 01:11:51.380 align:middle line:84% so the difference between the Euclidean setting 01:11:51.380 --> 01:11:54.140 align:middle line:90% and the finite-field setting. 01:11:54.140 --> 01:11:56.432 align:middle line:90% OK. 01:11:56.432 --> 01:11:58.990 align:middle line:84% OK, and now, once we know this, we are really 01:11:58.990 --> 01:12:02.270 align:middle line:84% done, because using Balog-Szemeredi-Gowers 01:12:02.270 --> 01:12:05.810 align:middle line:84% three times, we can assume that x is a Cartesian product. 01:12:05.810 --> 01:12:08.630 align:middle line:84% And we just need to know that the prediction in a given 01:12:08.630 --> 01:12:11.410 align:middle line:84% direction grows, and it is given by this. 01:12:11.410 --> 01:12:42.430 align:middle line:90% 01:12:42.430 --> 01:12:45.680 align:middle line:90% And then we apply double star. 01:12:45.680 --> 01:13:08.640 align:middle line:90% 01:13:08.640 --> 01:13:10.780 align:middle line:84% OK, so since there are five minutes left, 01:13:10.780 --> 01:13:14.000 align:middle line:84% let me summarize what's been going on. 01:13:14.000 --> 01:13:17.380 align:middle line:84% So what is the idea, the general idea of the proof, 01:13:17.380 --> 01:13:20.480 align:middle line:84% both in the finite-field setting and the Euclidean setting, 01:13:20.480 --> 01:13:24.640 align:middle line:84% first, we work with the set in one dimension, 01:13:24.640 --> 01:13:27.500 align:middle line:84% and shows that if the set is not already everything, 01:13:27.500 --> 01:13:29.960 align:middle line:84% then it's expanded by some polynomial. 01:13:29.960 --> 01:13:34.200 align:middle line:84% Then we simplify the polynomial using Plunnecke-Ruzsa. 01:13:34.200 --> 01:13:35.480 align:middle line:90% Then we iterate. 01:13:35.480 --> 01:13:38.440 align:middle line:84% In order to iterate, one has to be much, much, much more careful 01:13:38.440 --> 01:13:40.520 align:middle line:84% in the Euclidean setting in order 01:13:40.520 --> 01:13:43.000 align:middle line:84% to show that the hypothesis of the expansion 01:13:43.000 --> 01:13:46.680 align:middle line:84% are satisfied when we apply the growth once. 01:13:46.680 --> 01:13:48.060 align:middle line:90% So we can iterate. 01:13:48.060 --> 01:13:49.860 align:middle line:90% But eventually, we can iterate. 01:13:49.860 --> 01:13:52.600 align:middle line:84% That means that we can expand the given set A to almost 01:13:52.600 --> 01:13:54.850 align:middle line:90% everything by some polynomial. 01:13:54.850 --> 01:13:58.570 align:middle line:84% Once we can expand by polynomial to almost everything, 01:13:58.570 --> 01:14:00.690 align:middle line:90% we get this growth. 01:14:00.690 --> 01:14:02.810 align:middle line:84% It is still of some product type, 01:14:02.810 --> 01:14:06.690 align:middle line:84% but the difference is that x now can be much bigger than A. 01:14:06.690 --> 01:14:08.890 align:middle line:84% And the fact that x is much bigger than A is not 01:14:08.890 --> 01:14:11.330 align:middle line:84% an issue, because instead of working with A, 01:14:11.330 --> 01:14:15.250 align:middle line:84% we work with a polynomial applied to A, which has 01:14:15.250 --> 01:14:18.090 align:middle line:90% size bigger than the size of x. 01:14:18.090 --> 01:14:21.330 align:middle line:84% And finally, we want to project something which is not 01:14:21.330 --> 01:14:24.250 align:middle line:84% a product, but by looking at three projections 01:14:24.250 --> 01:14:27.370 align:middle line:84% to begin with, which are far away from each other, 01:14:27.370 --> 01:14:28.930 align:middle line:84% and applying Balog-Szemeredi-Gowers 01:14:28.930 --> 01:14:32.970 align:middle line:84% many times, we can go back to this setting. 01:14:32.970 --> 01:14:35.070 align:middle line:84% So that's the short version of the proof. 01:14:35.070 --> 01:14:37.470 align:middle line:84% And they are, again, in the Euclidean setting, 01:14:37.470 --> 01:14:42.370 align:middle line:84% one has to be careful to show that the projections satisfy 01:14:42.370 --> 01:14:49.630 align:middle line:84% the assumptions needed to apply this fact. 01:14:49.630 --> 01:14:52.570 align:middle line:84% So really, you need at least some concentration 01:14:52.570 --> 01:14:56.930 align:middle line:84% on x, because again, if x was a small interval, then 01:14:56.930 --> 01:15:01.320 align:middle line:84% this never grows for any A. So one needs some assumption on x. 01:15:01.320 --> 01:15:03.070 align:middle line:84% So one needs to prove something like that. 01:15:03.070 --> 01:15:08.650 align:middle line:84% But OK, one can prove it, and then eventually, one uses this. 01:15:08.650 --> 01:15:10.110 align:middle line:90% Any final questions? 01:15:10.110 --> 01:15:15.478 align:middle line:90% 01:15:15.478 --> 01:15:18.040 align:middle line:84% AUDIENCE: How many times have we used BSG? 01:15:18.040 --> 01:15:22.890 align:middle line:90% 01:15:22.890 --> 01:15:25.130 align:middle line:84% PABLO SHMERKIN: So either four or six. 01:15:25.130 --> 01:15:27.630 align:middle line:90% I'm not completely sure. 01:15:27.630 --> 01:15:28.130 align:middle line:90% No. 01:15:28.130 --> 01:15:30.213 align:middle line:84% I think the first time, we already have a product. 01:15:30.213 --> 01:15:36.770 align:middle line:84% Yeah, I think once, for A plus little aA, and then three times 01:15:36.770 --> 01:15:38.000 align:middle line:90% to reduce to this setting. 01:15:38.000 --> 01:15:41.250 align:middle line:90% 01:15:41.250 --> 01:15:45.570 align:middle line:84% AUDIENCE: Is it for when we just want the projection of x 01:15:45.570 --> 01:15:47.550 align:middle line:90% itself is big in one direction? 01:15:47.550 --> 01:15:53.478 align:middle line:84% Or it's like for the proof that's [INAUDIBLE]? 01:15:53.478 --> 01:15:54.520 align:middle line:90% PABLO SHMERKIN: Yeah, OK. 01:15:54.520 --> 01:15:57.880 align:middle line:84% So in the finite-field setting, as you are suggesting, 01:15:57.880 --> 01:16:00.660 align:middle line:84% it's much easier to prove that the projection of x itself 01:16:00.660 --> 01:16:03.660 align:middle line:84% is big than proving it for dense subsets. 01:16:03.660 --> 01:16:06.580 align:middle line:84% But in the Euclidean case, even if your goal 01:16:06.580 --> 01:16:08.740 align:middle line:84% is proving that the projection of x is large, 01:16:08.740 --> 01:16:10.820 align:middle line:84% and you don't care about dense subsets, 01:16:10.820 --> 01:16:13.580 align:middle line:84% you are still forced to care about dense subsets 01:16:13.580 --> 01:16:15.620 align:middle line:84% because you need the dense subsets to prove 01:16:15.620 --> 01:16:19.820 align:middle line:84% the non-concentration assumption when you iterate. 01:16:19.820 --> 01:16:23.740 align:middle line:84% So how did we prove that A plus little aA satisfies the stronger 01:16:23.740 --> 01:16:26.100 align:middle line:90% non-concentration assumption? 01:16:26.100 --> 01:16:29.340 align:middle line:84% By proving that the projection of dense subsets of A times A 01:16:29.340 --> 01:16:31.140 align:middle line:90% are large. 01:16:31.140 --> 01:16:34.220 align:middle line:84% So even if your ultimate goal doesn't 01:16:34.220 --> 01:16:35.880 align:middle line:84% involve looking at dense subsets, 01:16:35.880 --> 01:16:39.300 align:middle line:84% you still need to look at dense subsets along the proof. 01:16:39.300 --> 01:16:41.420 align:middle line:90% And you cannot avoid that. 01:16:41.420 --> 01:16:44.380 align:middle line:84% Or maybe you can, but at least with this method of proof, 01:16:44.380 --> 01:16:45.680 align:middle line:90% you cannot avoid that. 01:16:45.680 --> 01:16:52.207 align:middle line:90% 01:16:52.207 --> 01:16:52.870 align:middle line:90% Yes. 01:16:52.870 --> 01:16:54.495 align:middle line:84% AUDIENCE: Can you say a little bit more 01:16:54.495 --> 01:16:57.150 align:middle line:84% about how to handle the issue that the uniform subset 01:16:57.150 --> 01:17:01.435 align:middle line:84% at different scales might be different? 01:17:01.435 --> 01:17:02.310 align:middle line:90% PABLO SHMERKIN: Yeah. 01:17:02.310 --> 01:17:06.050 align:middle line:84% So unfortunately, I didn't realize this issue would arise. 01:17:06.050 --> 01:17:08.020 align:middle line:90% So I didn't check carefully. 01:17:08.020 --> 01:17:08.790 align:middle line:90% AUDIENCE: Oh, OK. 01:17:08.790 --> 01:17:10.310 align:middle line:90% That's fine. 01:17:10.310 --> 01:17:13.420 align:middle line:84% PABLO SHMERKIN: But yeah, it's a very good question. 01:17:13.420 --> 01:17:23.790 align:middle line:90% 01:17:23.790 --> 01:17:24.420 align:middle line:90% Yeah, OK. 01:17:24.420 --> 01:17:26.710 align:middle line:84% So I'm not exactly sure, but let me 01:17:26.710 --> 01:17:29.410 align:middle line:84% say something that might work in this case, 01:17:29.410 --> 01:17:31.590 align:middle line:84% but it certainly works in other cases. 01:17:31.590 --> 01:17:36.350 align:middle line:84% So I explained an idea superficially 01:17:36.350 --> 01:17:38.250 align:middle line:90% of first getting an A prime. 01:17:38.250 --> 01:17:40.390 align:middle line:84% And if that A prime doesn't exhaust A, 01:17:40.390 --> 01:17:42.950 align:middle line:84% we take it away, and find another A prime. 01:17:42.950 --> 01:17:45.310 align:middle line:84% One can do something similar for uniform sets. 01:17:45.310 --> 01:17:48.580 align:middle line:84% So rather than taking one uniform subset, 01:17:48.580 --> 01:17:52.480 align:middle line:84% so one is given a set, and one knows nothing about that set. 01:17:52.480 --> 01:17:55.840 align:middle line:84% And we can nearly exhaust that set 01:17:55.840 --> 01:17:58.760 align:middle line:84% by a finite union of uniform subsets. 01:17:58.760 --> 01:18:00.660 align:middle line:84% So one finds a dense uniform set, 01:18:00.660 --> 01:18:03.640 align:middle line:84% takes it away, finds another uniform set, takes it away. 01:18:03.640 --> 01:18:06.160 align:middle line:84% And in this way, we essentially cover the set 01:18:06.160 --> 01:18:09.600 align:middle line:84% with a very small error by uniform subsets. 01:18:09.600 --> 01:18:11.760 align:middle line:84% So this is one way to get around the issue 01:18:11.760 --> 01:18:15.128 align:middle line:84% that one uniform subset itself is maybe too sparse. 01:18:15.128 --> 01:18:17.670 align:middle line:84% And then a different scale is different, things could happen. 01:18:17.670 --> 01:18:22.040 align:middle line:90% 01:18:22.040 --> 01:18:24.100 align:middle line:84% So I think that could be one way. 01:18:24.100 --> 01:18:26.180 align:middle line:84% I don't think that's what we do in the paper. 01:18:26.180 --> 01:18:31.920 align:middle line:84% But I think that's one thing you could do to bypass this issue. 01:18:31.920 --> 01:18:34.560 align:middle line:84% AUDIENCE: You said last time that there was possibly a way 01:18:34.560 --> 01:18:38.160 align:middle line:90% to simplify this [INAUDIBLE]. 01:18:38.160 --> 01:18:39.960 align:middle line:84% PABLO SHMERKIN: We are not sure yet. 01:18:39.960 --> 01:18:41.720 align:middle line:90% We are hopeful. 01:18:41.720 --> 01:18:44.160 align:middle line:90% Yes. 01:18:44.160 --> 01:18:47.530 align:middle line:90% Yeah, again. 01:18:47.530 --> 01:18:52.210 align:middle line:84% So Bourgain's original proof is different in the sense 01:18:52.210 --> 01:18:57.250 align:middle line:84% that he didn't construct an expanding polynomial at all. 01:18:57.250 --> 01:19:03.250 align:middle line:84% But it is similar in the sense that he proves this, and then 01:19:03.250 --> 01:19:08.130 align:middle line:84% uses the many Balog-Szemeredi-Gowers to get 01:19:08.130 --> 01:19:09.530 align:middle line:90% the full statement. 01:19:09.530 --> 01:19:11.810 align:middle line:84% But to prove this, he did something different. 01:19:11.810 --> 01:19:14.050 align:middle line:90% But OK. 01:19:14.050 --> 01:19:16.990 align:middle line:84% Other claim is more difficult than what I explained. 01:19:16.990 --> 01:19:20.690 align:middle line:90% But I don't know. 01:19:20.690 --> 01:19:24.450 align:middle line:84% Yeah, now, we have this idea from last week 01:19:24.450 --> 01:19:27.610 align:middle line:84% that, yeah, might be a-- so basically, it 01:19:27.610 --> 01:19:29.810 align:middle line:84% would be a much easier way of doing this. 01:19:29.810 --> 01:19:31.395 align:middle line:84% To go from this to the full statement, 01:19:31.395 --> 01:19:32.770 align:middle line:84% the only way we know how to do it 01:19:32.770 --> 01:19:38.860 align:middle line:84% is how Bourgain did it, by doing the many Balog-Szemeredi-Gowers. 01:19:38.860 --> 01:19:45.000 align:middle line:90%