WEBVTT 00:00:17.880 --> 00:00:20.820 YUFEI ZHAO: Today we want to look at the sum product 00:00:20.820 --> 00:00:21.522 problem. 00:00:21.522 --> 00:00:22.980 So for the past few lectures, we've 00:00:22.980 --> 00:00:25.260 been discussing the structure of sets 00:00:25.260 --> 00:00:28.380 under the addition operation. 00:00:28.380 --> 00:00:31.050 Today we're going to throw in one extra operation, so 00:00:31.050 --> 00:00:33.310 multiplication, and understand how 00:00:33.310 --> 00:00:37.920 sets behave under both addition and multiplication. 00:00:37.920 --> 00:00:43.380 And the basic problem here is, can it be the case that A 00:00:43.380 --> 00:00:49.980 plus A, A times A, which is, analogously, 00:00:49.980 --> 00:00:55.860 the set of all pairwise products of elements from A-- 00:00:55.860 --> 00:01:05.760 can these two sets be simultaneously small, 00:01:05.760 --> 00:01:08.390 that is, the same for some single A? 00:01:11.710 --> 00:01:17.710 Can we have it so that A plus A and A times A 00:01:17.710 --> 00:01:20.690 are simultaneously small? 00:01:20.690 --> 00:01:24.470 For example, it's easy to make one of them small. 00:01:24.470 --> 00:01:29.210 We've seen examples where if you take A to be an arithmetic 00:01:29.210 --> 00:01:32.570 progression, then A plus A is more or less as 00:01:32.570 --> 00:01:34.400 small as it gets. 00:01:34.400 --> 00:01:39.710 But for such an example, you see A times A is pretty large. 00:01:39.710 --> 00:01:42.620 It's actually not so clear how to prove how large it 00:01:42.620 --> 00:01:43.820 Is. 00:01:43.820 --> 00:01:45.320 And there are some very nice proofs. 00:01:45.320 --> 00:01:48.620 And this problem has actually been more or less pinned down. 00:01:48.620 --> 00:01:54.350 But the short version is that A times A has size close 00:01:54.350 --> 00:01:55.790 to its maximum possible. 00:01:59.600 --> 00:02:05.690 So it turns out the size of A times A is almost quadratic. 00:02:05.690 --> 00:02:08.009 So this number is actually now known fairly precisely. 00:02:08.009 --> 00:02:12.410 So this problem of determining the size of A times 00:02:12.410 --> 00:02:17.690 A for the interval 1 through N is known as the Erdos 00:02:17.690 --> 00:02:19.160 multiplication table problem. 00:02:30.190 --> 00:02:32.910 So if you take an N by N multiplication table, 00:02:32.910 --> 00:02:35.920 how many numbers do you see in the table? 00:02:35.920 --> 00:02:39.570 So that turns out to be sub-quadratic, but not too 00:02:39.570 --> 00:02:42.170 sub-quadratic. 00:02:42.170 --> 00:02:46.130 So this problem has been more or less solved by Kevin Ford. 00:02:46.130 --> 00:02:48.840 And we now know a fairly precise expression, 00:02:48.840 --> 00:02:50.840 but I don't want to focus on that. 00:02:50.840 --> 00:02:54.200 That's not the topic of today's lecture. 00:02:54.200 --> 00:02:55.700 This is just an example. 00:02:55.700 --> 00:02:57.950 Alternatively, you can take A times A 00:02:57.950 --> 00:03:02.400 to be quite small by taking A to be a geometric progression. 00:03:02.400 --> 00:03:04.820 Then it's not too hard to convince yourself 00:03:04.820 --> 00:03:08.390 that A plus A must be fairly large in that case. 00:03:08.390 --> 00:03:10.310 And the geometric progression doesn't 00:03:10.310 --> 00:03:14.630 have so much additive structure, so A plus A will be large. 00:03:14.630 --> 00:03:20.380 So can you make A plus A and A times A simultaneously small? 00:03:20.380 --> 00:03:24.470 So there's this conjecture that the answer is no. 00:03:24.470 --> 00:03:27.480 And this is a famous conjecture in this area, known 00:03:27.480 --> 00:03:36.400 as the Erdos similarity conjecture on the sum product 00:03:36.400 --> 00:03:40.810 problem, which states that for all finite sets 00:03:40.810 --> 00:03:48.730 of real numbers, either A plus A or A times A 00:03:48.730 --> 00:03:53.035 has to be close to quadratic size. 00:04:00.300 --> 00:04:01.800 So that's the conjecture. 00:04:01.800 --> 00:04:03.810 It's still very much open. 00:04:03.810 --> 00:04:05.760 Today I want to show you some progress 00:04:05.760 --> 00:04:08.810 towards this conjecture via some partial results. 00:04:08.810 --> 00:04:12.780 And it will use a nice combination of tools 00:04:12.780 --> 00:04:16.519 from graph theory and incidence geometry, 00:04:16.519 --> 00:04:18.810 so it nicely ties in together many of the things 00:04:18.810 --> 00:04:22.019 that we've seen in this course so far. 00:04:22.019 --> 00:04:25.870 So Erdos and Szemeredi proved some bound, 00:04:25.870 --> 00:04:30.610 which is like 1 plus c for some constant c. 00:04:30.610 --> 00:04:35.387 Today we'll show some bounds for somewhat better c's. 00:04:35.387 --> 00:04:35.970 So you'll see. 00:04:39.350 --> 00:04:41.480 The first tool that I want to introduce 00:04:41.480 --> 00:04:45.150 is a result from graph theory known as the "crossing number 00:04:45.150 --> 00:04:46.312 inequality." 00:04:56.660 --> 00:04:58.340 So you know that planar graphs are 00:04:58.340 --> 00:05:00.100 graphs where you can draw on the planes 00:05:00.100 --> 00:05:02.230 so that the edges do not cross. 00:05:02.230 --> 00:05:05.000 And there are some famous examples of non-planar graphs, 00:05:05.000 --> 00:05:09.070 like K5 and K 3, 3. 00:05:09.070 --> 00:05:11.560 But you can ask a more quantitative question. 00:05:11.560 --> 00:05:14.320 If I give you a graph, how many crossings 00:05:14.320 --> 00:05:18.280 must you have in every drawing of this graph? 00:05:18.280 --> 00:05:19.840 And the crossing number inequality 00:05:19.840 --> 00:05:22.990 provides some estimate for such a quantity. 00:05:22.990 --> 00:05:28.600 So given the graph G, denoted by cr, so crossing of G, 00:05:28.600 --> 00:05:40.130 to be the minimum number of crossings in a planar 00:05:40.130 --> 00:05:46.570 drawing of G. There is a bit of subtlety 00:05:46.570 --> 00:05:50.470 here, where by a planar drawing, do I mean using line segments 00:05:50.470 --> 00:05:52.090 or do I mean using curves? 00:05:52.090 --> 00:05:55.990 It's actually not clear how it affects this quantity here. 00:05:55.990 --> 00:05:57.520 That's a very subtle issue. 00:05:57.520 --> 00:06:00.490 So for planar graphs, there's a famous result 00:06:00.490 --> 00:06:02.800 that more or less says if a planar graph can 00:06:02.800 --> 00:06:04.270 be drawn using continuous curves, 00:06:04.270 --> 00:06:06.830 then it can be drawn using straight lines. 00:06:06.830 --> 00:06:09.352 But the minimum number of crossings, 00:06:09.352 --> 00:06:10.810 the two different ways of drawings, 00:06:10.810 --> 00:06:13.400 they might end up with different crossing numbers. 00:06:13.400 --> 00:06:15.230 But for the purpose of today's lecture, 00:06:15.230 --> 00:06:18.010 we'll use a more general notion, although it doesn't actually 00:06:18.010 --> 00:06:19.840 matter for today which one we'll use-- 00:06:19.840 --> 00:06:21.745 so planar drawing using curves. 00:06:25.870 --> 00:06:28.810 Draw the graph where edges are continuous curves. 00:06:28.810 --> 00:06:30.460 How many crossings do you get? 00:06:30.460 --> 00:06:35.450 The crossing is a pair of edges that cross. 00:06:35.450 --> 00:06:38.260 You can ask-- it's just a cross over point that can-- 00:06:38.260 --> 00:06:39.035 it doesn't matter. 00:06:39.035 --> 00:06:40.660 So there are many different subtle ways 00:06:40.660 --> 00:06:42.040 of defining these things. 00:06:42.040 --> 00:06:44.830 They won't really come up for today's lecture. 00:06:44.830 --> 00:06:49.510 The crossing number inequality is a result 00:06:49.510 --> 00:06:54.010 from the '80s, which give you a lower-bound estimate 00:06:54.010 --> 00:06:56.530 on the number of crossings. 00:06:56.530 --> 00:07:06.470 If G is a graph with enough edges-- 00:07:06.470 --> 00:07:08.220 the number of edges is, let's say, 00:07:08.220 --> 00:07:10.790 at least four times the number of vertices-- 00:07:13.530 --> 00:07:21.220 then the number of crossings of every drawing of G 00:07:21.220 --> 00:07:24.970 is at least the number of edges cubed 00:07:24.970 --> 00:07:28.810 divided by the number of vertices squared. 00:07:28.810 --> 00:07:32.530 And there's an extra constant factor, which is some constant. 00:07:39.380 --> 00:07:43.570 So the constant does not depend on the graph. 00:07:43.570 --> 00:07:45.760 In particular, if it has a lot of edges, 00:07:45.760 --> 00:07:50.815 then every drawing of G must have a lot of crossings. 00:07:50.815 --> 00:07:52.300 So the crossing number inequality 00:07:52.300 --> 00:07:56.080 was proved by two separate independent works, 00:07:56.080 --> 00:07:59.680 one by Ajtai, Chvatal, Newborn, Szemeredi and the 00:07:59.680 --> 00:08:02.470 other by Tom Leighton, our very own Tom Leighton. 00:08:06.950 --> 00:08:11.090 So let me first give you some consequences of this theorem, 00:08:11.090 --> 00:08:12.690 just for illustration. 00:08:12.690 --> 00:08:17.930 So if you have an n-vertex graph with a quadratic number 00:08:17.930 --> 00:08:26.460 of edges, then how many crossings must you have? 00:08:26.460 --> 00:08:29.070 You plug in these parameters into the theorem. 00:08:29.070 --> 00:08:36.270 See that it has necessarily n to the 4th crossings. 00:08:39.100 --> 00:08:42.020 But if you just draw the graph in some arbitrary way, 00:08:42.020 --> 00:08:44.570 you have at most n to the 4 crossings, 00:08:44.570 --> 00:08:49.215 because a crossing involves four points. 00:08:49.215 --> 00:08:51.090 So when you have a quadratic number of edges, 00:08:51.090 --> 00:08:54.780 you must get basically the maximum number of crossings. 00:08:54.780 --> 00:08:58.163 The leading constant term factor is an interesting problem, 00:08:58.163 --> 00:08:59.580 which we're not going to get into. 00:09:02.540 --> 00:09:06.810 Let's prove the crossing number inequality. 00:09:06.810 --> 00:09:11.145 First, the base case of the crossing number inequalities 00:09:11.145 --> 00:09:14.065 is when you can draw a graph with no crossings. 00:09:14.065 --> 00:09:16.420 And those are planar graphs. 00:09:16.420 --> 00:09:29.130 So for every connected planar graph, 00:09:29.130 --> 00:09:31.770 if it has at least one cycle-- and you'll see why in a second, 00:09:31.770 --> 00:09:33.000 why I say this-- 00:09:33.000 --> 00:09:43.380 if with at least one cycle, so that's not a tree, 00:09:43.380 --> 00:09:48.150 we must have that 3 times the number of faces 00:09:48.150 --> 00:09:53.380 is at most 2 times the number of edges. 00:09:53.380 --> 00:09:55.770 So here, we're going to use the key tool 00:09:55.770 --> 00:10:01.380 being Euler's formula, which we all 00:10:01.380 --> 00:10:05.310 know as the number of vertices minus the number of edges 00:10:05.310 --> 00:10:09.080 plus the number of faces equals to 2. 00:10:09.080 --> 00:10:12.470 We're here for face, because I draw a planar graph, 00:10:12.470 --> 00:10:18.040 and so I count the faces. 00:10:18.040 --> 00:10:21.520 Here there are two faces, outer face, inner face, count edges 00:10:21.520 --> 00:10:25.380 and vertices, so you have Euler's formula up there. 00:10:25.380 --> 00:10:28.390 And plug in Euler's formula for a planar graph 00:10:28.390 --> 00:10:32.740 with at least one cycle, so we can obtain this consequence 00:10:32.740 --> 00:10:42.030 over here, because every face is adjacent to at least 00:10:42.030 --> 00:10:45.330 three edges. 00:10:45.330 --> 00:10:47.510 If you go around the face, you see 00:10:47.510 --> 00:10:56.350 these three edges, and every edge is counted exactly twice, 00:10:56.350 --> 00:11:02.320 is adjacent to exactly two faces. 00:11:02.320 --> 00:11:04.785 So you do the double counting, you get that inequality up 00:11:04.785 --> 00:11:05.285 there. 00:11:07.920 --> 00:11:15.910 So plugging these two into Euler gets you that inequality up 00:11:15.910 --> 00:11:16.410 there. 00:11:23.730 --> 00:11:31.290 Plugging these two into Euler, we get that the number of edges 00:11:31.290 --> 00:11:34.870 is almost 3 times the number of vertices minus 6. 00:11:41.662 --> 00:11:43.120 So for this leaves that inequality, 00:11:43.120 --> 00:11:48.760 but plug it into Euler, plug in this into Euler, you get this. 00:11:48.760 --> 00:11:52.300 So we have that the number of edges 00:11:52.300 --> 00:11:57.670 is at most 3 times the number of vertices for every graph G. 00:11:57.670 --> 00:12:02.230 So here, we require that the graph is planar 00:12:02.230 --> 00:12:07.205 and has at least one cycle, but even if we drop the condition 00:12:07.205 --> 00:12:08.830 that it has at least one cycle but just 00:12:08.830 --> 00:12:15.760 require that it's planar, every planar graph G 00:12:15.760 --> 00:12:18.645 satisfies this inequality over here. 00:12:18.645 --> 00:12:20.020 So in other words, you might have 00:12:20.020 --> 00:12:24.100 heard before, in a planar graph, the average degree of a vertex 00:12:24.100 --> 00:12:25.150 is almost 6. 00:12:28.460 --> 00:12:34.790 So in particular, the crossing number of a graph G 00:12:34.790 --> 00:12:40.430 is positive if the number of edges 00:12:40.430 --> 00:12:42.380 exceeds 3 times the number of vertices. 00:12:44.915 --> 00:12:46.810 It's not planar, so it has at least one 00:12:46.810 --> 00:12:49.300 crossing every drawing. 00:12:49.300 --> 00:13:04.730 And by deleting an edge from each crossing, 00:13:04.730 --> 00:13:05.900 we get a planar graph. 00:13:11.360 --> 00:13:13.040 You draw the graph. 00:13:13.040 --> 00:13:14.040 You have some crossings. 00:13:14.040 --> 00:13:17.230 You get rid of an edge associated with each drawing. 00:13:17.230 --> 00:13:19.450 Then you get a planar graph. 00:13:19.450 --> 00:13:21.610 If you look at this inequality and you 00:13:21.610 --> 00:13:24.460 account for the number of edges that you deleted, 00:13:24.460 --> 00:13:28.740 we obtain then the inequality that the number 00:13:28.740 --> 00:13:31.470 of edges minus the number of crossings 00:13:31.470 --> 00:13:34.710 is at least 3 times the number of vertices. 00:13:43.130 --> 00:13:50.820 So we obtain the inequality that the lower 00:13:50.820 --> 00:13:56.640 bounds in number of crossings as the number of edges 00:13:56.640 --> 00:14:01.110 minus 3 times the number of vertices, this one. 00:14:06.820 --> 00:14:10.490 So that's some lower bound on the crossing number. 00:14:10.490 --> 00:14:12.650 It's not quite the bound that we have over there. 00:14:12.650 --> 00:14:14.990 And in fact, if you take a graph with a quadratic number 00:14:14.990 --> 00:14:17.840 of edges, this bound here only gives you 00:14:17.840 --> 00:14:20.810 quadratic lower bound on the crossing number, some 00:14:20.810 --> 00:14:21.480 lower bound. 00:14:21.480 --> 00:14:22.860 But it's not a great lower bound. 00:14:22.860 --> 00:14:24.750 And we would like to do better. 00:14:24.750 --> 00:14:28.710 So here's a trick that is a very nice trick, 00:14:28.710 --> 00:14:33.410 where we're going to use this inequality to upgrade it 00:14:33.410 --> 00:14:36.170 to a much better inequality, bootstrap it 00:14:36.170 --> 00:14:38.430 to a much tighter inequality. 00:14:38.430 --> 00:14:40.910 So this involves the use of the probabilistic method. 00:14:43.870 --> 00:14:48.410 Let me denote by p some number between 0 and 1, 00:14:48.410 --> 00:14:49.760 to be decided later. 00:14:54.080 --> 00:14:57.320 And starting with a graph G, let's 00:14:57.320 --> 00:15:05.820 let G prime, with vertices and edges being V prime and E 00:15:05.820 --> 00:15:16.300 prime, be obtained from G by randomly deleting some 00:15:16.300 --> 00:15:19.690 of the vertices, or rather randomly 00:15:19.690 --> 00:15:29.600 keeping each vertex with probability p, 00:15:29.600 --> 00:15:35.931 independently for each of these vertices. 00:15:35.931 --> 00:15:37.940 So you have some graph G. 00:15:37.940 --> 00:15:41.330 I keep each vertex with probability p. 00:15:41.330 --> 00:15:43.640 And I delete the remaining vertices. 00:15:43.640 --> 00:15:45.800 And I get a smaller graph. 00:15:45.800 --> 00:15:49.230 I get some induced subgraph. 00:15:49.230 --> 00:15:51.440 And I would like to know what can we 00:15:51.440 --> 00:15:55.910 say about the crossing number of the smaller graph in comparison 00:15:55.910 --> 00:16:00.240 to the crossing number of the original graph? 00:16:00.240 --> 00:16:03.860 For the smaller graph, because it's still a planar graph 00:16:03.860 --> 00:16:05.770 so G prime-- 00:16:05.770 --> 00:16:07.180 so it's still a graph. 00:16:07.180 --> 00:16:09.180 It's not a planar graph, but it's still a graph, 00:16:09.180 --> 00:16:14.380 so G prime still satisfies this inequality up here. 00:16:17.600 --> 00:16:21.950 So G prime still satisfies that the number of crossings 00:16:21.950 --> 00:16:24.270 in every drawing of G prime is at least 00:16:24.270 --> 00:16:28.500 the number of edges of G prime minus 3 times the number 00:16:28.500 --> 00:16:30.520 of vertices of G prime. 00:16:33.560 --> 00:16:38.390 But note that G prime is a random graph. 00:16:38.390 --> 00:16:40.400 G was fixed, given. 00:16:40.400 --> 00:16:44.100 G prime is a random graph. 00:16:44.100 --> 00:16:46.520 So let's evaluate the expectation 00:16:46.520 --> 00:16:50.270 of both quantities, left-hand side and right-hand side. 00:16:53.730 --> 00:16:56.280 If this inequality is true for every G prime, 00:16:56.280 --> 00:16:59.370 the same inequality must be true in expectation. 00:17:09.369 --> 00:17:13.510 Now what do we know about all the expectations of each 00:17:13.510 --> 00:17:17.230 of these quantities? 00:17:17.230 --> 00:17:19.940 The number of vertices in expectation-- 00:17:19.940 --> 00:17:21.260 that's pretty easy. 00:17:21.260 --> 00:17:28.780 So this one here is p times the original number of vertices. 00:17:28.780 --> 00:17:30.970 The number of edges is also pretty easy. 00:17:30.970 --> 00:17:34.150 Each edge is kept if both endpoints are kept. 00:17:34.150 --> 00:17:38.620 So this expectation on the number of edges remaining 00:17:38.620 --> 00:17:43.640 is also pretty easy to determine. 00:17:43.640 --> 00:17:49.070 The crossing number of the new graph-- 00:17:49.070 --> 00:17:51.830 that I have to be a little bit more careful of, 00:17:51.830 --> 00:17:54.380 because when you look at the smaller graph, 00:17:54.380 --> 00:17:56.780 maybe there's a different way to draw it 00:17:56.780 --> 00:18:00.320 that's not just deleting the sum of the vertices 00:18:00.320 --> 00:18:02.070 from the original graph. 00:18:02.070 --> 00:18:03.560 So even though the original graph 00:18:03.560 --> 00:18:06.170 might have a lot of crossings, when you go to a subgraph, 00:18:06.170 --> 00:18:09.070 maybe there's a better way to draw it. 00:18:09.070 --> 00:18:11.320 But we just need an inequality in the right direction. 00:18:11.320 --> 00:18:13.110 So we are still OK. 00:18:13.110 --> 00:18:16.170 And I claim that the crossing number of G prime 00:18:16.170 --> 00:18:19.740 is in expectation at most p to be 4th 00:18:19.740 --> 00:18:23.190 times the crossing number of G. Because if you 00:18:23.190 --> 00:18:32.360 keep the same drawing, then the expected number of crossings 00:18:32.360 --> 00:18:34.470 that are kept-- 00:18:34.470 --> 00:18:38.850 each crossing is kept if all four of its end points 00:18:38.850 --> 00:18:40.770 are kept. 00:18:40.770 --> 00:18:44.830 So each crossing is kept with probability p to the 4th. 00:18:44.830 --> 00:18:47.320 So you can draw it in expectation 00:18:47.320 --> 00:18:48.845 with this many crossings. 00:18:48.845 --> 00:18:49.720 Maybe it's much less. 00:18:49.720 --> 00:18:50.890 Maybe there's a better way to draw it, 00:18:50.890 --> 00:18:53.575 but you have an inequality going in the right direction. 00:19:01.500 --> 00:19:05.130 Looking at that inequality up there in yellow, 00:19:05.130 --> 00:19:08.710 we find that the crossing number of G 00:19:08.710 --> 00:19:17.960 is at least p to the minus 2 E minus 3p to the minus 3. 00:19:21.090 --> 00:19:25.690 And this is true for every value of p between 0 and 1. 00:19:25.690 --> 00:19:30.540 So now you pick a value of p that works most in your favor. 00:19:30.540 --> 00:19:33.690 And it turns out you should do this 00:19:33.690 --> 00:19:37.710 by setting these two equalities to be 00:19:37.710 --> 00:19:39.100 roughly equal to each other. 00:19:43.050 --> 00:19:57.090 So setting p between 0 and 1 so that 4 times the-- 00:19:57.090 --> 00:19:58.590 basically, set these two terms to be 00:19:58.590 --> 00:20:00.020 roughly equal to each other. 00:20:03.890 --> 00:20:09.320 And then we get that this quantity here 00:20:09.320 --> 00:20:13.910 is at least the claimed quantity, 00:20:13.910 --> 00:20:19.270 which is E cubed over V squared up 00:20:19.270 --> 00:20:24.140 to some constant factor, which I don't really care about. 00:20:24.140 --> 00:20:27.020 In order to set p, I have to be a little bit careful 00:20:27.020 --> 00:20:28.560 that p is between 0 and 1. 00:20:28.560 --> 00:20:30.500 If you set p to be 1.2, this whole argument 00:20:30.500 --> 00:20:33.060 doesn't make any sense. 00:20:33.060 --> 00:20:33.980 So this is OK. 00:20:36.820 --> 00:20:46.260 So we know p is at most one as long as E is at most 4p. 00:20:46.260 --> 00:20:49.470 I mean, the 4 here is not optimal, but if 4 were 2, 00:20:49.470 --> 00:20:50.550 then it's not true. 00:20:50.550 --> 00:20:54.482 So if E is 2V, you can have a planar graph, 00:20:54.482 --> 00:20:56.940 so you shouldn't have a lower bound on the crossing number. 00:20:59.550 --> 00:21:02.620 So this is the proof of the crossing number inequality. 00:21:02.620 --> 00:21:04.770 As I said, if you have lots of edges, 00:21:04.770 --> 00:21:08.960 then you must have lots of crossings. 00:21:08.960 --> 00:21:10.002 Any questions? 00:21:13.250 --> 00:21:15.200 So let's use the crossing number inequality 00:21:15.200 --> 00:21:19.220 to prove a fundamental result in incidence geometry. 00:21:26.820 --> 00:21:30.120 Incidence geometry is this area of discrete math 00:21:30.120 --> 00:21:33.510 that concerns fairly basic-sounding questions 00:21:33.510 --> 00:21:37.530 about incidences between, let's say, points and lines. 00:21:37.530 --> 00:21:39.730 And here's an example. 00:21:39.730 --> 00:21:52.530 So what's the maximum number of incidences between endpoints 00:21:52.530 --> 00:21:57.660 and end lines, where by "incidence" 00:21:57.660 --> 00:22:03.150 I mean if p-- so curly p-- is a set of points, 00:22:03.150 --> 00:22:09.750 and curly l is a set of lines, then I write I of p and l 00:22:09.750 --> 00:22:20.770 to be the number of pairs, one point, one line, such 00:22:20.770 --> 00:22:25.380 that the point lies on the line. 00:22:25.380 --> 00:22:29.440 So I'm counting incidences between points and lines. 00:22:29.440 --> 00:22:30.910 You can view this in many ways. 00:22:30.910 --> 00:22:34.000 You can view it as a bipartite graph between points and lines, 00:22:34.000 --> 00:22:40.000 and we're counting the number of edges in this bipartite graph. 00:22:40.000 --> 00:22:41.730 So I give you end points, end lines. 00:22:41.730 --> 00:22:45.350 What's the maximum number of incidences? 00:22:45.350 --> 00:22:47.670 It's not such an obvious question. 00:22:47.670 --> 00:22:52.560 So let's see how we can approach this question. 00:22:52.560 --> 00:22:58.280 But first, let me give you some easy bounds. 00:22:58.280 --> 00:23:02.480 So here's a trivial bound-- 00:23:07.680 --> 00:23:12.390 so here, I want to know if I give you some number of points, 00:23:12.390 --> 00:23:16.250 some number of lines, what's the maximum number of incidences. 00:23:16.250 --> 00:23:21.580 So a trivial bound is that the number of incidences 00:23:21.580 --> 00:23:26.970 is at most the product between the number of points 00:23:26.970 --> 00:23:30.120 and the number of lines. 00:23:30.120 --> 00:23:32.220 One point, one line, at most one incidence. 00:23:32.220 --> 00:23:34.750 So that's pretty trivial. 00:23:34.750 --> 00:23:36.260 We can do better. 00:23:36.260 --> 00:23:42.840 So we can do better because, well, you 00:23:42.840 --> 00:23:50.050 see, let's use this following fact, that every line-- 00:23:52.880 --> 00:24:02.735 so every pair of points determine at most one line. 00:24:02.735 --> 00:24:03.810 I have two points. 00:24:03.810 --> 00:24:08.780 There's at most one line that contains those two points. 00:24:08.780 --> 00:24:15.990 Using this fact, we see that the number of-- 00:24:15.990 --> 00:24:23.500 so let's count the number of triples involving 00:24:23.500 --> 00:24:34.590 two points and one line such that both points lie 00:24:34.590 --> 00:24:35.220 on the line. 00:24:39.730 --> 00:24:41.530 So how big can this set be? 00:24:41.530 --> 00:24:44.950 So let's try to count it in two different ways. 00:24:44.950 --> 00:24:48.900 On one hand, this quantity is at most 00:24:48.900 --> 00:24:52.270 the number of points squared, because if I give you 00:24:52.270 --> 00:24:56.700 two points, then they determine this line-- 00:24:56.700 --> 00:25:01.580 so at most the number of points squared. 00:25:01.580 --> 00:25:09.390 But on the other hand, we see that if I give you a line, 00:25:09.390 --> 00:25:12.090 I just need to count now the number of-- 00:25:15.268 --> 00:25:17.560 let me also require that these two points are distinct. 00:25:17.560 --> 00:25:20.860 So if I give you a line, I now need 00:25:20.860 --> 00:25:26.990 to count the number of pairs of points on this line. 00:25:26.990 --> 00:25:36.750 So I can enumerate over lines and count line 00:25:36.750 --> 00:25:42.270 by line how many pairs of points are on that line. 00:25:42.270 --> 00:25:45.790 So I get this quantity over here. 00:25:45.790 --> 00:25:48.930 On each line, I have that contribution. 00:25:48.930 --> 00:25:55.100 And now, using Cauchy-Schwartz inequality, 00:25:55.100 --> 00:26:01.130 we find that this squared term is at least 00:26:01.130 --> 00:26:11.200 the number of incidences divided by the number of lines. 00:26:11.200 --> 00:26:13.660 And the remaining minus 1 term contributes just 00:26:13.660 --> 00:26:16.134 to the number of incidences. 00:26:20.090 --> 00:26:22.370 So the first is by Cauchy-Schwartz. 00:26:26.690 --> 00:26:30.450 So putting these two inequalities together, 00:26:30.450 --> 00:26:34.920 we get some upper bound on the number of incidences. 00:26:34.920 --> 00:26:37.990 If you have to invert this inequality, 00:26:37.990 --> 00:26:42.810 you will get that the number of incidences between points 00:26:42.810 --> 00:26:50.360 and lines is upper bounded by the number of points 00:26:50.360 --> 00:26:53.930 times the number of lines raised to power 00:26:53.930 --> 00:27:00.450 1/2 plus the number of lines. 00:27:00.450 --> 00:27:03.290 So that's what you get from this inequality over here. 00:27:06.150 --> 00:27:08.720 By considering point-line duality-- 00:27:08.720 --> 00:27:12.450 so whenever you have this kind of setup involving points 00:27:12.450 --> 00:27:15.870 and lines, you can take the projected duality 00:27:15.870 --> 00:27:18.570 and transform the configuration into-- 00:27:18.570 --> 00:27:21.150 lines into points and points into lines, and the incidences 00:27:21.150 --> 00:27:22.560 are preserved. 00:27:22.560 --> 00:27:25.630 So I also have an inequality. 00:27:25.630 --> 00:27:29.550 By duality-- I also have an inequality 00:27:29.550 --> 00:27:32.760 where I switch the roles of points and lines. 00:27:38.610 --> 00:27:40.890 So I is already the numbers. 00:27:40.890 --> 00:27:44.470 I don't need to put an extra absolute value sign. 00:27:44.470 --> 00:27:46.590 So the number of points and lines 00:27:46.590 --> 00:27:50.130 is upper bounded by the number of lines 00:27:50.130 --> 00:27:52.740 times the square root of a number of points 00:27:52.740 --> 00:27:58.200 plus an extra term, just in case there are very few lines. 00:28:02.035 --> 00:28:03.910 So these are the bounds that you have so far. 00:28:03.910 --> 00:28:06.160 And the only thing that we have used so far 00:28:06.160 --> 00:28:09.340 is the fact that every two points determine at most one 00:28:09.340 --> 00:28:12.760 line, and every two lines meet at at most one point. 00:28:15.290 --> 00:28:17.200 So these are the bounds that we get. 00:28:17.200 --> 00:28:23.720 And in particular, for end points and end lines, 00:28:23.720 --> 00:28:26.530 we get the number of incidences is-- 00:28:26.530 --> 00:28:29.420 they go off n to the 3/2. 00:28:34.440 --> 00:28:36.930 This should remind you of something we've done before. 00:28:40.300 --> 00:28:43.560 So in the first part of this course, 00:28:43.560 --> 00:28:50.500 when we were looking at extremal numbers, where did 3/2 come up? 00:28:50.500 --> 00:28:52.420 AUDIENCE: [INAUDIBLE] like C4? 00:28:52.420 --> 00:28:53.980 YUFEI ZHAO: C4, yeah. 00:28:53.980 --> 00:29:00.450 So if you compare this quantity to the extremal number of C4, 00:29:00.450 --> 00:29:04.690 it's also n to the 3/2. 00:29:04.690 --> 00:29:07.750 And in fact, the proof is exactly the same. 00:29:07.750 --> 00:29:12.110 All we're using here is that the incidence graph is C4-free 00:29:12.110 --> 00:29:17.700 So in fact, this is an argument about C4-free graphs. 00:29:17.700 --> 00:29:21.060 So this fact here, every two points determine at most one 00:29:21.060 --> 00:29:25.080 line, is saying that if you look at the incidence graph, 00:29:25.080 --> 00:29:28.000 there's no C4. 00:29:28.000 --> 00:29:30.670 That's all we're using for now. 00:29:30.670 --> 00:29:31.712 Any questions? 00:29:35.410 --> 00:29:37.000 So is this the truth? 00:29:37.000 --> 00:29:39.640 Now, back when we were discussing the extremal number 00:29:39.640 --> 00:29:43.650 for C4-free graphs, we saw that, in fact, this 00:29:43.650 --> 00:29:45.240 is the correct order. 00:29:45.240 --> 00:29:46.740 And what was the construction there? 00:29:53.550 --> 00:29:56.790 So the construction also came from incidences, 00:29:56.790 --> 00:30:01.290 but incidences of taking all lines 00:30:01.290 --> 00:30:07.950 and points in the finite field plain, Fq squared. 00:30:07.950 --> 00:30:10.740 If you look at all the lines and all the points 00:30:10.740 --> 00:30:13.740 in a finite field plain, then you 00:30:13.740 --> 00:30:18.750 get the correct lower bound for C4. 00:30:18.750 --> 00:30:23.490 But now we are actually working in the real plane, 00:30:23.490 --> 00:30:28.560 so it turns out that the answer is different when you're not 00:30:28.560 --> 00:30:30.120 working the finite field. 00:30:30.120 --> 00:30:33.615 We're going to be using the topology of the real plane. 00:30:33.615 --> 00:30:35.740 And we're going to come up with a different answer. 00:30:35.740 --> 00:30:41.070 So it turns out that the truth for the number 00:30:41.070 --> 00:30:44.210 of maximum number of incidences in the plane, 00:30:44.210 --> 00:30:50.020 for points and lines in the real plane, is not exponent 3/2, 00:30:50.020 --> 00:30:53.770 but turns out to be 4/3. 00:30:53.770 --> 00:30:56.710 And this is a consequence of an important result 00:30:56.710 --> 00:30:58.950 in incidence geometry, a fundamental result, 00:30:58.950 --> 00:31:00.800 known as the Szemeredi-Trotter theorem. 00:31:07.080 --> 00:31:11.350 So the Szemeredi-Trotter theorem says 00:31:11.350 --> 00:31:17.050 that the number of incidences between points and lines 00:31:17.050 --> 00:31:19.720 is upper bounded by this function where 00:31:19.720 --> 00:31:22.740 you look at the number of points times the number of lines, 00:31:22.740 --> 00:31:34.250 and each raised to power 2/3 and plus some additional terms, 00:31:34.250 --> 00:31:38.360 just in case there are many more lines compared to points or way 00:31:38.360 --> 00:31:41.170 more points compared to lines. 00:31:41.170 --> 00:31:44.070 So that's the Szemeredi-Trotter theorem. 00:31:44.070 --> 00:31:52.190 And as a corollary, you see that n points, n lines give you 00:31:52.190 --> 00:31:58.930 at most n to the 4/3 incidences, in contrast 00:31:58.930 --> 00:32:05.950 to the setting of the finite field plain, where you can 00:32:05.950 --> 00:32:08.150 get n to the 3/2 incidences. 00:32:08.150 --> 00:32:10.870 So somehow, we have to use the topology 00:32:10.870 --> 00:32:13.280 of the real plane for this one. 00:32:13.280 --> 00:32:15.370 And I want to show you a proof-- 00:32:15.370 --> 00:32:17.080 turns out not the original proof, 00:32:17.080 --> 00:32:19.490 but it's a proof that uses the crossing number 00:32:19.490 --> 00:32:23.525 inequality to prove Szemeredi-Trotter theorem. 00:32:23.525 --> 00:32:25.150 You see, in crossing number inequality, 00:32:25.150 --> 00:32:29.290 we are using the topology of the real plane. 00:32:29.290 --> 00:32:31.657 Where? 00:32:31.657 --> 00:32:32.740 AUDIENCE: Euler's formula. 00:32:32.740 --> 00:32:34.198 YUFEI ZHAO: Euler's formula, right. 00:32:34.198 --> 00:32:36.020 So the very beginning, Euler's formula 00:32:36.020 --> 00:32:40.290 has to do with the topology of the real plane. 00:32:40.290 --> 00:32:43.790 Now, this bound turns out to be tight. 00:32:43.790 --> 00:32:45.530 So let me give you an example showing 00:32:45.530 --> 00:32:49.880 that the 4/3 exponent is tight. 00:32:49.880 --> 00:32:55.340 And the example is, if you take p 00:32:55.340 --> 00:33:06.420 to be this rectangular grid of points, 00:33:06.420 --> 00:33:09.750 and L to be a set of lines-- so I'm 00:33:09.750 --> 00:33:13.710 going to write the lines by their equation, 00:33:13.710 --> 00:33:17.730 where the slope is an integer from 1 through k 00:33:17.730 --> 00:33:21.090 and the y-intercept is an integer from 1 00:33:21.090 --> 00:33:23.610 through k squared. 00:33:23.610 --> 00:33:30.230 And you see here that every line in L 00:33:30.230 --> 00:33:42.130 contains exactly k points from P. So we got in total k 00:33:42.130 --> 00:33:51.040 to the 4th incidences, which is on the order of n to the 4/3. 00:33:53.840 --> 00:33:55.740 So n to the 4/3 third is the right answer. 00:33:59.513 --> 00:34:01.930 Now let me show you how to prove Szemeredi-Trotter theorem 00:34:01.930 --> 00:34:03.980 from the crossing number inequality. 00:34:03.980 --> 00:34:06.700 It turns out to be a very neat application that's 00:34:06.700 --> 00:34:10.610 almost a direct consequence once you set up the right graph. 00:34:10.610 --> 00:34:15.550 And the idea is that we are going to draw a graph based 00:34:15.550 --> 00:34:19.570 on our incidence configuration. 00:34:19.570 --> 00:34:26.900 So first, just to clean things up a little bit, 00:34:26.900 --> 00:34:41.830 let's get rid of lines in L with 1 or 0 points in P. 00:34:41.830 --> 00:34:45.219 So this operation doesn't affect the bounds. 00:34:45.219 --> 00:34:46.810 So you can check. 00:34:46.810 --> 00:34:50.170 These lines don't contribute much to the incidence bound, 00:34:50.170 --> 00:34:53.260 and only contributes to this plus L. 00:34:53.260 --> 00:34:55.600 So you can get rid of such lines. 00:34:55.600 --> 00:35:03.830 So let's assume that every line in L 00:35:03.830 --> 00:35:14.390 contains at least two points from P. 00:35:14.390 --> 00:35:19.010 And let's draw a graph based on this incidence structure. 00:35:19.010 --> 00:35:19.970 So if I have-- 00:35:27.850 --> 00:35:36.980 so suppose these are my points and lines. 00:35:36.980 --> 00:35:39.560 I'll just draw a graph where I keep 00:35:39.560 --> 00:35:47.560 the points as the vertices, and I put in an edge. 00:35:47.560 --> 00:35:55.900 It's a finite edge that connects two adjacent points 00:35:55.900 --> 00:35:56.620 on the same line. 00:36:02.467 --> 00:36:03.300 So I get some graph. 00:36:09.480 --> 00:36:12.938 Let me make this graph a bit more interesting. 00:36:23.680 --> 00:36:25.080 So I get some graph. 00:36:25.080 --> 00:36:31.340 And how many crossings, at most, does this graph have? 00:36:31.340 --> 00:36:42.030 So the number of crossings of G is at most 00:36:42.030 --> 00:36:45.920 the number of lines squared, because a crossing 00:36:45.920 --> 00:36:47.030 comes from two lines. 00:36:49.662 --> 00:36:50.870 So here, you have a crossing. 00:36:50.870 --> 00:36:52.448 A crossing comes from two lines. 00:36:52.448 --> 00:36:54.740 Number of crossings is at most number of lines squared. 00:36:57.380 --> 00:36:59.500 On the other hand, we can give a lower bound 00:36:59.500 --> 00:37:04.820 to the number of crossings from the crossing number inequality. 00:37:04.820 --> 00:37:07.470 And to do that, I want to estimate the number of edges. 00:37:07.470 --> 00:37:09.770 And this is the reason why I assume every line contains 00:37:09.770 --> 00:37:16.580 at least two points from P, because a line with now k 00:37:16.580 --> 00:37:23.720 incidences gives k minus 1 edges. 00:37:26.690 --> 00:37:32.300 And if k is at least 2, then k minus 1 is at least k over 2, 00:37:32.300 --> 00:37:33.410 let's say. 00:37:33.410 --> 00:37:35.990 I don't care about constant factors. 00:37:35.990 --> 00:37:41.270 So by crossing number inequality, 00:37:41.270 --> 00:37:46.130 the number of crossings of G is at least 00:37:46.130 --> 00:37:50.210 the number of edges cubed over the number of vertices 00:37:50.210 --> 00:38:00.320 squared, which is at least the number of incidences 00:38:00.320 --> 00:38:05.980 of this configuration cubed over the number of points squared. 00:38:05.980 --> 00:38:08.540 Actually, number of vertices is the number of points. 00:38:08.540 --> 00:38:12.050 And number of edges, by this argument here, 00:38:12.050 --> 00:38:15.428 is on the same order as the number of incidences. 00:38:18.500 --> 00:38:24.470 Putting these two facts together, we see-- 00:38:24.470 --> 00:38:29.510 there was one extra hypothesis in crossing number inequality. 00:38:29.510 --> 00:38:32.780 Provided that this hypothesis holds, 00:38:32.780 --> 00:38:37.610 which is that the number of incidences 00:38:37.610 --> 00:38:48.720 is at least 8 times the number of points, 00:38:48.720 --> 00:38:52.200 so that the original hypothesis holds. 00:38:54.810 --> 00:38:58.080 So putting everything together, and rearranging 00:38:58.080 --> 00:39:02.190 all of these terms, and using upper and lower bounds 00:39:02.190 --> 00:39:08.460 on the crossing number, we find that the number of incidences 00:39:08.460 --> 00:39:10.980 is upper bounded by-- 00:39:10.980 --> 00:39:22.750 the main term you see is just coming from these two, 00:39:22.750 --> 00:39:27.640 but there are a few other terms that we should put in, just 00:39:27.640 --> 00:39:31.170 in case this hypothesis is violated, 00:39:31.170 --> 00:39:35.080 and also to take care of this assumption over here, 00:39:35.080 --> 00:39:39.660 so adding a couple of linear terms corresponding 00:39:39.660 --> 00:39:43.200 to the number of points and the number of lines. 00:39:43.200 --> 00:39:46.677 If this hypothesis is violated, then the inequality 00:39:46.677 --> 00:39:47.260 is still true. 00:39:51.650 --> 00:39:55.430 So this proves the crossing numbers inequality. 00:39:55.430 --> 00:39:56.638 Any questions? 00:40:00.790 --> 00:40:06.450 So we've done these two very neat results. 00:40:06.450 --> 00:40:09.760 The question is, what do they have to do with the sum product 00:40:09.760 --> 00:40:11.790 problem? 00:40:11.790 --> 00:40:15.390 So I want to show you how you can give some lower bound 00:40:15.390 --> 00:40:20.010 on the sum product problem using Szemeredi-Trotter theorem. 00:40:22.650 --> 00:40:25.750 So it turns out that the sum product problem is intimately 00:40:25.750 --> 00:40:28.780 related to incidence geometry. 00:40:28.780 --> 00:40:32.500 And the reason-- you'll see in a second precisely why they're 00:40:32.500 --> 00:40:36.640 related, but roughly speaking, when you have addition 00:40:36.640 --> 00:40:39.580 and multiplication, they're are kind of 00:40:39.580 --> 00:40:43.090 like taking slope and y-intercept 00:40:43.090 --> 00:40:45.020 of an equation of a line. 00:40:45.020 --> 00:40:47.500 So there are two operations that are involved. 00:40:47.500 --> 00:40:52.660 So turns out, many incidence geometry problems can be set up 00:40:52.660 --> 00:40:53.830 and a way-- 00:40:53.830 --> 00:40:55.630 so many sum product problems can be set up 00:40:55.630 --> 00:40:58.800 in a way that involves incidence geometry. 00:40:58.800 --> 00:41:04.330 And a very short and clever lower bound to the sum product 00:41:04.330 --> 00:41:10.270 problem was proved by Elekes in the late '90s. 00:41:18.930 --> 00:41:25.470 So he showed the bound that if you have a subset of finite, 00:41:25.470 --> 00:41:31.870 subset of reals, then the sum set size times the product set 00:41:31.870 --> 00:41:35.665 size is at least A to the 5/2. 00:41:39.150 --> 00:41:46.995 As a corollary, one of these two must be fairly large. 00:41:46.995 --> 00:41:53.560 The max of the sum set size and the product set size 00:41:53.560 --> 00:41:57.495 is at least a to the 5/4. 00:42:06.030 --> 00:42:07.440 Let me show you the proof. 00:42:07.440 --> 00:42:11.040 I'm going to construct a set of points and a set of lines 00:42:11.040 --> 00:42:16.490 based on the set A. And the set of points in R2 00:42:16.490 --> 00:42:22.700 is going to be pairs x comma y, where the horizontal coordinate 00:42:22.700 --> 00:42:26.740 lies in the sum set, A plus A, and the vertical coordinate 00:42:26.740 --> 00:42:38.140 lies in the product set, A times A. And a set of lines 00:42:38.140 --> 00:42:40.350 is going to be these lines-- 00:42:40.350 --> 00:42:52.350 y equals to a times x minus a prime, where a and a prime 00:42:52.350 --> 00:42:56.810 lie in A. 00:42:56.810 --> 00:43:00.840 So these are some points and some lines. 00:43:00.840 --> 00:43:07.270 And I want to show you that they must have many incidences. 00:43:07.270 --> 00:43:09.180 So what are the incidences? 00:43:09.180 --> 00:43:17.080 So note that the line y equals to a times x minus a prime-- 00:43:17.080 --> 00:43:27.510 it contains the points a prime plus b and ab, 00:43:27.510 --> 00:43:34.230 which lies in P for all b in A. You plug it in. 00:43:34.230 --> 00:43:39.743 If you plug in a prime plus b into here, you get ab. 00:43:39.743 --> 00:43:44.300 And this point lies in P, because the first coordinate 00:43:44.300 --> 00:43:45.920 is the sum set. 00:43:45.920 --> 00:43:49.550 The second coordinate lies in the product set. 00:43:49.550 --> 00:43:57.570 So each line in L contains many incidences. 00:43:57.570 --> 00:44:01.960 So each line in L contains a incidents. 00:44:01.960 --> 00:44:17.610 So this line, each line in L contains a incidences. 00:44:17.610 --> 00:44:23.490 Also, we can easily compute the number of lines 00:44:23.490 --> 00:44:26.490 and the number of points. 00:44:26.490 --> 00:44:30.870 The number of points is A plus A size 00:44:30.870 --> 00:44:36.540 times the size of A times A. And the number of lines 00:44:36.540 --> 00:44:41.100 is just the size of A squared. 00:44:41.100 --> 00:44:52.541 So by Szemeredi-Trotter, we find that the number of incidences 00:44:52.541 --> 00:44:58.250 is lower bounded by noting this fact here. 00:44:58.250 --> 00:44:59.880 We have many incidences. 00:44:59.880 --> 00:45:07.120 So the number of lines, each line contributes a incidences. 00:45:07.120 --> 00:45:09.100 But we also have an upper bound coming 00:45:09.100 --> 00:45:11.496 from the Szemeredi-Trotter theorem. 00:45:11.496 --> 00:45:18.100 So plugging in the upper bound, we find that you have-- 00:45:18.100 --> 00:45:20.080 so now I'm just directly plugging 00:45:20.080 --> 00:45:22.895 in the statement of Szemeredi-Trotter. 00:45:26.460 --> 00:45:28.317 The main term is the first term. 00:45:28.317 --> 00:45:30.150 You should still check the latter two terms, 00:45:30.150 --> 00:45:31.650 but the main term is the first term. 00:45:34.190 --> 00:45:39.540 So plugging in the values for P and L, 00:45:39.540 --> 00:45:55.790 we find this is the case, plus some additional terms, 00:45:55.790 --> 00:45:58.910 which you can check are dominated by the first term. 00:45:58.910 --> 00:46:00.770 So let me just do a big O over there. 00:46:03.530 --> 00:46:06.810 Now you put left and right together, 00:46:06.810 --> 00:46:11.070 and we could obtain some lower bound 00:46:11.070 --> 00:46:15.090 on the product of the sizes of the sum set and the product 00:46:15.090 --> 00:46:18.320 set, thereby yielding allocations. 00:46:23.710 --> 00:46:26.460 So this is some lower bound on the sum product problem. 00:46:26.460 --> 00:46:30.400 And you see, we went through the crossing number inequality 00:46:30.400 --> 00:46:32.875 to prove Szemeredi-Trotter, a basic result 00:46:32.875 --> 00:46:34.480 in incidence geometry. 00:46:34.480 --> 00:46:39.580 And viewing sum product as an incidence geometry problem, one 00:46:39.580 --> 00:46:43.850 can obtain this lower bound over here. 00:46:43.850 --> 00:46:44.878 Any questions? 00:46:47.950 --> 00:46:51.640 I want to show you a different proof that was found later, 00:46:51.640 --> 00:46:54.550 that gives an improvement. 00:46:54.550 --> 00:47:01.670 And there's a question, can you do better than 5/4? 00:47:01.670 --> 00:47:06.490 So it turns out that there was a very nice result of Solymosi 00:47:06.490 --> 00:47:09.914 sometime later that gives you an improvement. 00:47:16.210 --> 00:47:21.730 Solymosi proved in 2009 that if A 00:47:21.730 --> 00:47:29.030 is a subset of positive reals, then the size of A times 00:47:29.030 --> 00:47:33.070 A multiplied by the size of A plus A squared 00:47:33.070 --> 00:47:37.540 is at least size of A to the 4th divided 00:47:37.540 --> 00:47:45.610 by 4 ceiling log of the size of A, where the log is base 2. 00:47:45.610 --> 00:47:48.600 So don't worry about the specific constants. 00:47:51.240 --> 00:47:54.640 A being in the positive reals is no big deal, 00:47:54.640 --> 00:47:58.630 because you can always separate A as positive and negative 00:47:58.630 --> 00:48:00.790 and analyze each part separately. 00:48:00.790 --> 00:48:05.260 So as a corollary to Solymosi's theorem, 00:48:05.260 --> 00:48:12.730 we obtain that for A, a subset of the reals, the sum 00:48:12.730 --> 00:48:17.110 set and the product set, at least one of them 00:48:17.110 --> 00:48:24.990 must have size at least A raised to 4/3 00:48:24.990 --> 00:48:34.820 divided by 2 times log base 2 size of A raised to 1/3 third. 00:48:34.820 --> 00:48:42.210 So basically, A to the 4/3 minus little one in the exponent, 00:48:42.210 --> 00:48:43.370 so better than before. 00:48:43.370 --> 00:48:44.440 And this is a new bound. 00:48:47.580 --> 00:48:52.200 I want to note that in this formulation, where 00:48:52.200 --> 00:48:58.320 we are looking at lower bounding this quantity over here, 00:48:58.320 --> 00:49:06.160 this is tied up to logarithmic factors, 00:49:06.160 --> 00:49:11.710 by considering A to be just the interval from 1 to n. 00:49:11.710 --> 00:49:14.140 If A is the interval from 1 to n, 00:49:14.140 --> 00:49:17.230 then the left-hand side, A plus A, is around size n. 00:49:17.230 --> 00:49:18.660 So you have n squared. 00:49:18.660 --> 00:49:20.260 And A times A is also, I mentioned, 00:49:20.260 --> 00:49:23.800 around size n squared. 00:49:23.800 --> 00:49:26.170 So this inequality here is tight. 00:49:26.170 --> 00:49:28.820 The consequence is not tight, but the first inequality 00:49:28.820 --> 00:49:29.320 is tight. 00:49:34.400 --> 00:49:36.260 So in the remainder of today's lecture, 00:49:36.260 --> 00:49:39.960 I want to show you how to prove Solymosi's lower bound. 00:49:39.960 --> 00:49:43.550 And it has some similarities to the one 00:49:43.550 --> 00:49:50.210 that we've seen, because it also looks at some geometric aspects 00:49:50.210 --> 00:49:52.610 of the sum product problem. 00:49:52.610 --> 00:49:57.950 But it doesn't use the exact tools that we've seen earlier. 00:49:57.950 --> 00:50:00.590 It does use some tools that were related 00:50:00.590 --> 00:50:04.520 to the lecture from Monday. 00:50:04.520 --> 00:50:07.070 So last time, we discussed this thing 00:50:07.070 --> 00:50:09.830 called the "additive energy." 00:50:09.830 --> 00:50:12.560 You can come up with a similar notion for the multiplication 00:50:12.560 --> 00:50:24.220 operation, so the "multiplicative energy," 00:50:24.220 --> 00:50:31.426 which we'll denote by E sub, with the multiplication symbol, 00:50:31.426 --> 00:50:35.650 A. So the multiplicative energy is like the additive energy, 00:50:35.650 --> 00:50:38.028 except that instead of doing addition, 00:50:38.028 --> 00:50:39.820 we're going to do a multiplication instead. 00:50:39.820 --> 00:50:45.650 So one way to define it is the number of quadruples such 00:50:45.650 --> 00:50:55.530 that there exists some real lambda such that a, comma, 00:50:55.530 --> 00:50:58.680 b equals to lambda c, comma, d. 00:51:06.400 --> 00:51:08.540 So basically the same as additive energy, 00:51:08.540 --> 00:51:12.530 except that we're using multiplications instead. 00:51:12.530 --> 00:51:15.330 By the Cauchy-Schwartz inequality-- 00:51:15.330 --> 00:51:20.980 and this is a calculation we saw last time, as well-- 00:51:20.980 --> 00:51:27.440 we see that if you have a set with small product, then 00:51:27.440 --> 00:51:30.280 it must have high multiplicative energy. 00:51:30.280 --> 00:51:34.030 So last time, we saw small sum set implies high additive 00:51:34.030 --> 00:51:34.570 energy. 00:51:34.570 --> 00:51:38.560 Likewise, small product set implies high multiplicative 00:51:38.560 --> 00:51:39.570 energy. 00:51:39.570 --> 00:51:43.690 In particular, the multiplicative energy of A, 00:51:43.690 --> 00:51:50.440 you can rewrite it as sum over all elements 00:51:50.440 --> 00:51:55.750 x in the product set of the quantity, which tells you 00:51:55.750 --> 00:52:02.730 the number of ways to write x as a product, 00:52:02.730 --> 00:52:05.860 this number squared and then summed over all x. 00:52:05.860 --> 00:52:09.280 By Cauchy-Schwartz, we find that this quantity here is lower 00:52:09.280 --> 00:52:12.430 bounded by the size of A to the 4th divided 00:52:12.430 --> 00:52:20.200 by the size of A times A. So to prove Solymosi's theorem, 00:52:20.200 --> 00:52:26.710 we are going to actually prove a bound on the energy, 00:52:26.710 --> 00:52:28.680 instead of proving it on the set. 00:52:28.680 --> 00:52:30.460 We're going to prove it on the energy. 00:52:30.460 --> 00:52:42.480 So it suffices to show that the multiplicative energy is 00:52:42.480 --> 00:52:49.900 at most 4 times the sum set size times-- 00:52:49.900 --> 00:53:01.832 so let me divide the energy by log of A. 00:53:01.832 --> 00:53:04.580 So when you plug this into this inequality, 00:53:04.580 --> 00:53:05.520 it would imply that. 00:53:05.520 --> 00:53:09.590 So it remains to show this inequality 00:53:09.590 --> 00:53:12.170 over here upper bounding the multiplicative energy. 00:53:20.730 --> 00:53:22.710 There's an important idea that we're 00:53:22.710 --> 00:53:25.440 going to use here, which is also pretty common in analysis, 00:53:25.440 --> 00:53:32.850 is that instead of considering that energy sum here, 00:53:32.850 --> 00:53:35.880 we're going to consider a similar sum, 00:53:35.880 --> 00:53:40.110 except we're going to chop up the sum into pieces according 00:53:40.110 --> 00:53:44.600 to how big the terms are, so that we're only 00:53:44.600 --> 00:53:48.230 looking at contributions of comparable size. 00:53:48.230 --> 00:53:50.948 And so this is called a "dyadic decomposition." 00:54:02.420 --> 00:54:07.900 The idea is that we can write the multiplicative energy 00:54:07.900 --> 00:54:10.400 similar to above, but instead of summing over 00:54:10.400 --> 00:54:15.590 x in the product set, let me sum over s in the quotient set. 00:54:15.590 --> 00:54:21.670 So you can interpret what this quotient A is. 00:54:21.670 --> 00:54:27.140 This is the set of all A divided by B, where A and B are in A. A 00:54:27.140 --> 00:54:29.150 is a set of positive reals, so I don't need 00:54:29.150 --> 00:54:31.400 to worry about division by 0. 00:54:31.400 --> 00:54:36.770 So what remains, then, is the intersection 00:54:36.770 --> 00:54:41.700 of s times A and A squared. 00:54:41.700 --> 00:54:47.170 Remember, s times A is scaling each element of A by s. 00:54:47.170 --> 00:54:49.760 So we have this quantity over here. 00:54:49.760 --> 00:54:55.070 So I want to break up the sum into a bunch of smaller 00:54:55.070 --> 00:54:59.990 sums, where I want to break up the sum according 00:54:59.990 --> 00:55:05.570 to how big the terms are, so that inside each group, 00:55:05.570 --> 00:55:08.990 all the terms are roughly of the same size. 00:55:08.990 --> 00:55:11.480 And easiest way to do this is to chop them up 00:55:11.480 --> 00:55:19.400 into groups where everything inside the same collection 00:55:19.400 --> 00:55:21.540 differs by at most a factor of 2. 00:55:21.540 --> 00:55:25.100 So that's why it's called a dyadic decomposition, 00:55:25.100 --> 00:55:27.740 going from 0 to-- 00:55:27.740 --> 00:55:32.960 the maximum possible here is basically A. 00:55:32.960 --> 00:55:39.410 So let's look at i going from 0 to log base 2 of A. 00:55:39.410 --> 00:55:41.150 So this is the number of bins. 00:55:44.350 --> 00:55:49.810 And partition the sum into sub-sums 00:55:49.810 --> 00:55:54.190 where I'm looking at the i-th sub-sum consisting 00:55:54.190 --> 00:55:58.510 of contributions involving terms with size between 2 00:55:58.510 --> 00:56:01.700 to the i and 2 to the i plus 1. 00:56:09.530 --> 00:56:15.260 Break up the sum according to the sizes of the summands. 00:56:15.260 --> 00:56:18.020 By pigeonhole principle, one of these summands 00:56:18.020 --> 00:56:19.760 must be somewhat large. 00:56:23.520 --> 00:56:33.730 So by pigeonhole, there exists a k 00:56:33.730 --> 00:56:47.920 such that setting D to be the s such that that corresponds 00:56:47.920 --> 00:56:52.670 to the k-th term in the sum. 00:57:04.250 --> 00:57:17.790 So one has that this sum coming from just contributions from D 00:57:17.790 --> 00:57:19.260 is at least-- 00:57:23.760 --> 00:57:27.930 so it's at least the multiplicative energy 00:57:27.930 --> 00:57:29.760 divided by the number of bins. 00:57:36.170 --> 00:57:39.100 All of that many bins-- 00:57:39.100 --> 00:57:41.300 by pigeonhole, I can find one bin 00:57:41.300 --> 00:57:44.510 that's a pretty large contribution to the sum. 00:57:44.510 --> 00:57:52.910 And the right-hand side, we can upper bound each term over here 00:57:52.910 --> 00:57:57.470 by 2 to the 2k plus 2, and the number 00:57:57.470 --> 00:58:05.590 of terms as the size of D. Let me call the elements of D S1 00:58:05.590 --> 00:58:16.820 through Sm, where S1 through Sm are sorted in increasing order. 00:58:22.880 --> 00:58:26.960 Now let me draw you a picture of what's going on. 00:58:26.960 --> 00:58:40.050 Let's consider for each element of D, so for each i and m, 00:58:40.050 --> 00:58:48.230 let's consider the line given by the equation y equals to s 00:58:48.230 --> 00:58:51.960 sub i times x. 00:58:51.960 --> 00:58:56.290 Let me draw this picture where I'm 00:58:56.290 --> 00:59:01.260 looking at the positive quadrant, 00:59:01.260 --> 00:59:03.640 so I have a bunch of points in the positive quadrant. 00:59:10.180 --> 00:59:13.110 And specifically, I'm interested in these points whose 00:59:13.110 --> 00:59:18.205 coordinates, both coordinates are elements of A. 00:59:18.205 --> 00:59:26.580 And I want to consider lines through points of A, 00:59:26.580 --> 00:59:28.950 but I want to consider lines where 00:59:28.950 --> 00:59:34.380 it intersects this A cross A in the desired number of points. 00:59:37.030 --> 00:59:39.400 And we find those set, and then let's 00:59:39.400 --> 00:59:48.120 draw these lines over here, where this line here, L1 00:59:48.120 --> 00:59:53.640 has slope exactly S1, and L2, L3, and so on. 00:59:58.990 --> 01:00:03.820 I want to draw one more line, which is somewhat auxiliary, 01:00:03.820 --> 01:00:06.770 but just to make our life a bit easier. 01:00:06.770 --> 01:00:13.240 Finally, let's let L of m plus 1 be the vertical line, 01:00:13.240 --> 01:00:21.030 or rather be the vertical ray, which 01:00:21.030 --> 01:00:31.230 goes to the minimum element of A above Lm. 01:00:31.230 --> 01:00:34.660 So it's this line over here. 01:00:34.660 --> 01:00:36.370 That's Lm plus 1. 01:00:40.550 --> 01:00:44.920 So in A cross A, I draw a bunch of lines. 01:00:44.920 --> 01:00:46.650 So now all the lines-- 01:00:46.650 --> 01:00:50.700 so all these lines involve some point of A and the origin, 01:00:50.700 --> 01:00:51.930 but I don't draw all of them. 01:00:51.930 --> 01:00:54.600 I draw a select set of them. 01:00:54.600 --> 01:00:59.620 And what we said earlier says that the number of lines, 01:00:59.620 --> 01:01:02.880 the number of points on each of these strong lines, 01:01:02.880 --> 01:01:05.730 is roughly the same for each of these lines. 01:01:12.350 --> 01:01:17.240 Let's let capital L sub j denote the set 01:01:17.240 --> 01:01:24.350 of points in A cross A that lie on the j-th line. 01:01:32.320 --> 01:01:37.930 So that's L1, L2, and so on. 01:01:41.230 --> 01:01:49.420 I claim that if you look at two consecutive lines 01:01:49.420 --> 01:01:55.390 and look at the sum set of the points in A cross A 01:01:55.390 --> 01:02:00.500 that intersect, you're looking at two lines, 01:02:00.500 --> 01:02:02.550 and you're adding up points on those two lines. 01:02:02.550 --> 01:02:05.680 So you form a grid. 01:02:05.680 --> 01:02:13.210 So you end up forming this grid. 01:02:13.210 --> 01:02:15.940 And the number of points on this grid 01:02:15.940 --> 01:02:19.630 is precisely the product of these two point sets. 01:02:28.940 --> 01:02:42.650 Moreover, the sets Lj plus L sub j plus 1 01:02:42.650 --> 01:02:48.050 are disjoint for different j. 01:02:52.660 --> 01:02:56.440 And this is where we're using the geometry of the plane here. 01:02:56.440 --> 01:03:02.960 Because the sum of L1 and L2 lies in the span, 01:03:02.960 --> 01:03:06.470 the sum of L2 and L3 in a different span, 01:03:06.470 --> 01:03:09.370 so they cannot intersect. 01:03:09.370 --> 01:03:11.430 So they lie in-- 01:03:11.430 --> 01:03:38.370 so since they span disjoint regions, L1 plus L2 lies here, 01:03:38.370 --> 01:03:41.590 L2 plus L3 lies there, and so on. 01:03:41.590 --> 01:03:42.780 But they're all disjoint. 01:03:51.180 --> 01:03:53.700 Now let's put everything that we know together. 01:03:58.720 --> 01:04:06.550 Remember, the goal is to upper bound the multiplicative energy 01:04:06.550 --> 01:04:09.840 as a function of the sum set. 01:04:09.840 --> 01:04:12.300 So in other words, we want to lower bound the sum set. 01:04:12.300 --> 01:04:19.360 So I want to show you that this A plus A has a lot of elements. 01:04:19.360 --> 01:04:21.690 There's a lot of sums. 01:04:21.690 --> 01:04:25.360 And I have a bunch of disjoint contributions to these sums. 01:04:25.360 --> 01:04:28.590 So let's add up those disjoint contributions to the sums. 01:04:32.883 --> 01:04:38.220 You see that the size of A plus A squared 01:04:38.220 --> 01:04:42.990 is the same as the size of the product set A plus A. 01:04:42.990 --> 01:04:45.540 So this is Cartesian product. 01:04:45.540 --> 01:04:55.110 Here is-- this is a Cartesian product, 01:04:55.110 --> 01:04:58.740 in other words, the grid that is strong up there. 01:04:58.740 --> 01:05:01.150 I add this product to itself. 01:05:04.753 --> 01:05:06.170 So I should get the same set here. 01:05:09.660 --> 01:05:11.800 But how big is this sum set? 01:05:11.800 --> 01:05:15.290 That grid, that lattice grid added to itself, 01:05:15.290 --> 01:05:16.180 how big should it be? 01:05:16.180 --> 01:05:19.220 I want to lower bound the number of sums. 01:05:19.220 --> 01:05:23.030 And the key observation is up there. 01:05:23.030 --> 01:05:27.880 We can look at contributions coming from distinct spans. 01:05:27.880 --> 01:05:34.980 In particular, this sum here, so this sum set here, 01:05:34.980 --> 01:05:45.260 size is lower bounded by these distinct Lj plus L j plus 1's. 01:05:45.260 --> 01:05:46.210 I threw away a lot. 01:05:46.210 --> 01:05:49.430 I only keep the lines on the L's, and I only consider sums 01:05:49.430 --> 01:05:52.330 between consecutive L's. 01:05:52.330 --> 01:05:54.955 That should be a lower bound to the sum 01:05:54.955 --> 01:05:56.280 set of the grid with itself. 01:05:58.860 --> 01:06:03.425 But you see, and here, we're using these different-- 01:06:03.425 --> 01:06:08.460 for different j's, these contributions are destroyed. 01:06:08.460 --> 01:06:15.640 But by what we said up there, Lj plus L j plus 1 is a grid. 01:06:15.640 --> 01:06:20.950 So it has size Lj times L j plus 1. 01:06:23.900 --> 01:06:32.900 And the size of each Lj is at least 2 to the k. 01:06:32.900 --> 01:06:38.150 So the sum here is at least m times 2 to the 2k. 01:06:41.410 --> 01:06:48.340 But we saw over here that the energy lower bounds 01:06:48.340 --> 01:06:50.290 this 2 to the 2k. 01:06:50.290 --> 01:06:58.060 So we have a lower bound that is the multiplicative energy of A 01:06:58.060 --> 01:07:05.407 divided by 4 times the log base 2 of the size of A. 01:07:05.407 --> 01:07:07.490 So don't worry so much about the constant factors. 01:07:07.490 --> 01:07:13.000 That's just the order of magnitude that is important. 01:07:13.000 --> 01:07:14.680 And that's it. 01:07:14.680 --> 01:07:15.309 Yep. 01:07:15.309 --> 01:07:20.130 AUDIENCE: How do you know that the size of big L sub m plus 1? 01:07:20.130 --> 01:07:20.880 YUFEI ZHAO: Great. 01:07:20.880 --> 01:07:24.200 The question is, what do we know about the size of big L sub 01:07:24.200 --> 01:07:25.172 m plus 1? 01:07:25.172 --> 01:07:26.130 So that's a good point. 01:07:28.940 --> 01:07:30.543 The easiest answer is, if I don't 01:07:30.543 --> 01:07:31.960 care about these constant factors, 01:07:31.960 --> 01:07:34.660 I don't need to worry about it. 01:07:34.660 --> 01:07:40.060 You can think about what is the number of points 01:07:40.060 --> 01:07:47.430 on this line above that. 01:07:47.430 --> 01:07:53.110 It's essentially the number of elements of A above the biggest 01:07:53.110 --> 01:07:57.450 element of s m, above s m. 01:08:01.197 --> 01:08:03.123 It's a good question. 01:08:03.123 --> 01:08:04.790 I think we don't need to worry about it. 01:08:04.790 --> 01:08:08.935 I'm being slightly sloppy here. 01:08:08.935 --> 01:08:10.915 Yeah. 01:08:10.915 --> 01:08:16.890 AUDIENCE: [INAUDIBLE] 01:08:16.890 --> 01:08:18.390 YUFEI ZHAO: I think the question is, 01:08:18.390 --> 01:08:22.540 how do we know for j equals to m that you have this bound over 01:08:22.540 --> 01:08:23.040 here? 01:08:25.866 --> 01:08:33.770 AUDIENCE: [INAUDIBLE] 01:08:33.770 --> 01:08:34.520 YUFEI ZHAO: Great. 01:08:34.520 --> 01:08:35.260 So yes. 01:08:38.224 --> 01:08:48.877 AUDIENCE: [INAUDIBLE] 01:08:48.877 --> 01:08:50.710 YUFEI ZHAO: So there are some ways to do it. 01:08:50.710 --> 01:08:53.010 You can notice that the vertical line 01:08:53.010 --> 01:09:00.340 has at least as many points as the first slanted line. 01:09:03.420 --> 01:09:08.700 So details that you can work on. 01:09:08.700 --> 01:09:11.109 So this proves Solymosi's theorem, 01:09:11.109 --> 01:09:16.470 which gives you a lower bound on the sum set and the product 01:09:16.470 --> 01:09:19.600 set sizes and the maximum of those two. 01:09:19.600 --> 01:09:21.282 It's based on-- it's very short. 01:09:21.282 --> 01:09:21.990 It's very clever. 01:09:21.990 --> 01:09:24.540 It took a long time to find. 01:09:24.540 --> 01:09:28.649 And it gave a bound on the sum product 01:09:28.649 --> 01:09:32.370 problem of 4/3 that actually remained 01:09:32.370 --> 01:09:36.810 stuck for a very long time, until just 01:09:36.810 --> 01:09:48.420 fairly recently there was an improvement that gives-- 01:09:48.420 --> 01:09:52.350 so by Konyagin and Shkredov where 01:09:52.350 --> 01:09:58.260 they improved the Solymosi bound from 4/3 to 4/3 01:09:58.260 --> 01:10:00.570 plus some really small constant c. 01:10:04.300 --> 01:10:06.530 So it's some explicit constant. 01:10:06.530 --> 01:10:09.040 I think right now-- so that's being proved over time, 01:10:09.040 --> 01:10:12.780 but right now, I think c is around 1 over 1,000 01:10:12.780 --> 01:10:13.730 or a few thousand. 01:10:13.730 --> 01:10:17.600 So it's some small but explicit constant. 01:10:17.600 --> 01:10:22.170 It remains a major open problem to improve this bound 01:10:22.170 --> 01:10:25.730 and prove Erdos' similarity conjecture, 01:10:25.730 --> 01:10:31.070 that if you have n elements, then one of the sums 01:10:31.070 --> 01:10:33.830 or products must be nearly quadratic in size. 01:10:33.830 --> 01:10:36.650 And people generally believe that that's the case. 01:10:39.320 --> 01:10:40.468 Any questions? 01:10:47.950 --> 01:10:51.410 So this concludes all the topics I want to cover in this course. 01:10:51.410 --> 01:10:52.470 So we went a long way. 01:10:52.470 --> 01:10:54.110 And so the beginning of this course, 01:10:54.110 --> 01:10:56.860 we started with extremal graph theory, 01:10:56.860 --> 01:11:00.760 looking at the basic problem of if you have a graph that 01:11:00.760 --> 01:11:05.830 doesn't contain some subgraph, triangle, C4, what's 01:11:05.830 --> 01:11:08.440 the maximum number of edges. 01:11:08.440 --> 01:11:11.350 In fact, that showed up even today. 01:11:11.350 --> 01:11:13.270 And then we went down to other tools, 01:11:13.270 --> 01:11:15.640 like Szemeredi's regularity lemma 01:11:15.640 --> 01:11:18.250 that allows us to deduce important arithmetic 01:11:18.250 --> 01:11:20.710 consequences, such as Roth's theorem. 01:11:20.710 --> 01:11:22.210 It's also an extremal problem if you 01:11:22.210 --> 01:11:25.300 have a set without a three-term arithmetic progression, 01:11:25.300 --> 01:11:28.880 how many elements can it have? 01:11:28.880 --> 01:11:31.900 And so the important tool of Szemeredi's regularity lemma 01:11:31.900 --> 01:11:34.270 then later showed up in many different ways 01:11:34.270 --> 01:11:36.490 in this course, especially the message 01:11:36.490 --> 01:11:38.800 of Szemeredi's regularity lemma, that when 01:11:38.800 --> 01:11:41.290 you look at an object, it's important to decompose it 01:11:41.290 --> 01:11:44.800 into its structural component and its pseudo-random 01:11:44.800 --> 01:11:45.970 component. 01:11:45.970 --> 01:11:48.340 So this dichotomy, this interplay 01:11:48.340 --> 01:11:50.590 between structure and pseudo randomness, 01:11:50.590 --> 01:11:54.190 is a key theme throughout this course. 01:11:54.190 --> 01:11:56.170 And it showed up in some of the later topics 01:11:56.170 --> 01:11:59.920 as well, when we discussed spectral graph theory, 01:11:59.920 --> 01:12:03.250 quasi-randomness, graph limits, and also 01:12:03.250 --> 01:12:07.300 in the later Fourier analytic proof of Roth's theorem. 01:12:07.300 --> 01:12:09.490 All of these proofs, all of these techniques, 01:12:09.490 --> 01:12:12.250 involve some kind of interplay between structure 01:12:12.250 --> 01:12:15.990 and pseudo-randomness. 01:12:15.990 --> 01:12:17.700 In the past month or, so we've been 01:12:17.700 --> 01:12:21.240 looking at Freiman's theorem, this key result 01:12:21.240 --> 01:12:24.840 in additive combinatorics concerning the structure 01:12:24.840 --> 01:12:27.020 of sets under addition. 01:12:27.020 --> 01:12:30.450 And there, we also saw many different tools that came up, 01:12:30.450 --> 01:12:33.720 and also connections I mentioned a few lectures ago, 01:12:33.720 --> 01:12:35.490 connections to really important results 01:12:35.490 --> 01:12:38.220 in geometry to group theory. 01:12:38.220 --> 01:12:41.190 And it really extends all around. 01:12:41.190 --> 01:12:43.710 And a few takeaways from this course-- 01:12:43.710 --> 01:12:47.260 one of them is that graph theory, additive combinatorics, 01:12:47.260 --> 01:12:49.030 they are not isolated subjects. 01:12:49.030 --> 01:12:52.143 They're connected to a lot within mathematics. 01:12:52.143 --> 01:12:54.060 And that's one of the goals I want to show you 01:12:54.060 --> 01:12:57.600 in this course, is to show these connections 01:12:57.600 --> 01:13:01.170 throughout mathematics and some to analysis, to geometry, 01:13:01.170 --> 01:13:02.250 to topology. 01:13:02.250 --> 01:13:05.550 And even simple questions can lead 01:13:05.550 --> 01:13:08.790 to really deep mathematics. 01:13:08.790 --> 01:13:11.280 And some of them I try to show you, try to hint at you, 01:13:11.280 --> 01:13:13.980 or at least I mentioned throughout this course. 01:13:13.980 --> 01:13:20.220 And what we've seen so far is just the tip of the iceberg. 01:13:20.220 --> 01:13:23.430 And there is a lot of still extremely exciting work 01:13:23.430 --> 01:13:24.600 that's to be done. 01:13:24.600 --> 01:13:28.500 And I've also tried to emphasize many important open problems 01:13:28.500 --> 01:13:32.370 that have yet to be better understood. 01:13:32.370 --> 01:13:36.373 And I expect in some future iteration of this course, 01:13:36.373 --> 01:13:38.040 some of these problems will be resolved, 01:13:38.040 --> 01:13:41.460 and I can show the next generation of students 01:13:41.460 --> 01:13:43.740 in your seats some new techniques, new methods, 01:13:43.740 --> 01:13:44.752 and new theorems. 01:13:44.752 --> 01:13:46.210 And I expect that will be the case. 01:13:46.210 --> 01:13:47.670 This is a very exciting area. 01:13:47.670 --> 01:13:50.200 And it's an area that is very close to my heart. 01:13:50.200 --> 01:13:54.170 It's something that I've been thinking about since my PhD. 01:13:54.170 --> 01:13:56.730 The bulk of my research work revolves 01:13:56.730 --> 01:13:59.460 around better understanding connections between graph 01:13:59.460 --> 01:14:02.130 theory, on one hand, and additive combinatorics 01:14:02.130 --> 01:14:03.470 on the other hand. 01:14:03.470 --> 01:14:05.220 It's been really fun teaching this course, 01:14:05.220 --> 01:14:07.680 and happy to have all of you here. 01:14:07.680 --> 01:14:08.580 Thank you. 01:14:08.580 --> 01:14:11.630 [APPLAUSE]