WEBVTT 00:00:19.195 --> 00:00:19.820 YUFEI ZHAO: OK. 00:00:19.820 --> 00:00:22.760 I want to begin by giving some comments regarding 00:00:22.760 --> 00:00:23.865 the Wikipedia assignment. 00:00:23.865 --> 00:00:26.480 So I sent out an email about this last night. 00:00:26.480 --> 00:00:29.510 And so first of all, thank you for your contributions 00:00:29.510 --> 00:00:32.280 to this assignment, to Wikipedia. 00:00:32.280 --> 00:00:36.410 It plays a really important role in educating a wider audience 00:00:36.410 --> 00:00:39.260 what this subject is about because as many of you 00:00:39.260 --> 00:00:41.555 have experienced, the first time-- 00:00:41.555 --> 00:00:43.520 if you heard of some term, you have 00:00:43.520 --> 00:00:45.560 no idea what it is, you put it into Google, 00:00:45.560 --> 00:00:49.220 and often Wikipedia's entry is one of the top results that 00:00:49.220 --> 00:00:50.120 come up. 00:00:50.120 --> 00:00:52.700 And what gets written in there actually 00:00:52.700 --> 00:00:56.990 plays a fairly influential role in educating a broader audience 00:00:56.990 --> 00:00:59.520 about what this topic is about. 00:00:59.520 --> 00:01:04.370 And so I want to emphasize that this is not simply 00:01:04.370 --> 00:01:05.930 some homework assignment. 00:01:05.930 --> 00:01:08.300 It's something that is a real contribution. 00:01:08.300 --> 00:01:10.610 And it's something that contributes 00:01:10.610 --> 00:01:13.370 to the dissemination of knowledge. 00:01:13.370 --> 00:01:15.860 And for that, it is really important to do a good job, 00:01:15.860 --> 00:01:21.020 to do it right, to do it well, so that next time someone-- 00:01:21.020 --> 00:01:22.962 maybe even yourselves-- maybe you've 00:01:22.962 --> 00:01:24.920 forgotten what the subject is about and go back 00:01:24.920 --> 00:01:28.750 and you want to look it up again and remind yourself. 00:01:28.750 --> 00:01:32.420 You will have a useful resource to look into. 00:01:32.420 --> 00:01:35.450 But also let's say someone wants to find out what 00:01:35.450 --> 00:01:38.180 is external graph theory about? 00:01:38.180 --> 00:01:40.700 What is additive combinatorics about? 00:01:40.700 --> 00:01:43.190 You want them to land on the page that points you 00:01:43.190 --> 00:01:46.160 to the right type of places, that points you 00:01:46.160 --> 00:01:50.870 to useful resources, that opens doors so that you can explore 00:01:50.870 --> 00:01:51.680 further. 00:01:51.680 --> 00:01:55.480 And some of the contributions, indeed, 00:01:55.480 --> 00:01:57.170 serve you well in that purpose. 00:01:57.170 --> 00:01:59.600 It opens doors to many things. 00:01:59.600 --> 00:02:01.520 And part of the spirit of this assignment 00:02:01.520 --> 00:02:04.010 is for you to do your own research, 00:02:04.010 --> 00:02:07.940 do your own literature search, to learn more about a subject, 00:02:07.940 --> 00:02:11.060 more than what has been taught in these lectures 00:02:11.060 --> 00:02:13.770 so that you can write about it on Wikipedia. 00:02:13.770 --> 00:02:15.970 You can link to more references, you 00:02:15.970 --> 00:02:20.460 know, show the world what the subject is about. 00:02:20.460 --> 00:02:25.130 OK, continuing with our program, so we 00:02:25.130 --> 00:02:28.160 spent the past few lectures developing tools regarding 00:02:28.160 --> 00:02:31.220 the structure of set addition so that we 00:02:31.220 --> 00:02:33.050 can prove Freiman's theorem. 00:02:33.050 --> 00:02:35.630 So that's been our goal for the past few lectures. 00:02:35.630 --> 00:02:38.570 And today we'll finally prove Freiman's theorem. 00:02:38.570 --> 00:02:40.690 But let me first remind you the statement-- 00:02:40.690 --> 00:02:43.300 so in Freiman's theorem, we would like to show that if you 00:02:43.300 --> 00:02:44.580 are in a subset-- 00:02:44.580 --> 00:02:47.780 if you're in the integers, you have a set A that has bounded 00:02:47.780 --> 00:02:52.130 doubling-- doubling constant, constant k-- 00:02:52.130 --> 00:02:55.430 then the said must be contained in a small, generalized 00:02:55.430 --> 00:02:59.120 arithmetic progression, down to dimension and size, 00:02:59.120 --> 00:03:03.750 only a constant factor larger than a. 00:03:03.750 --> 00:03:08.190 We developed various tools the past three lectures building up 00:03:08.190 --> 00:03:09.750 to your intermediate results. 00:03:09.750 --> 00:03:12.000 But we also collected this very nice set of tools 00:03:12.000 --> 00:03:14.580 for proving Freiman's theorem. 00:03:14.580 --> 00:03:16.890 So let me review some of them, which 00:03:16.890 --> 00:03:19.890 we'll encounter again today. 00:03:19.890 --> 00:03:24.510 PlünneckeRuzsa inequality tells you that if you have a set with 00:03:24.510 --> 00:03:27.690 small doubling, then the further iterated sums are also 00:03:27.690 --> 00:03:28.860 controlled. 00:03:28.860 --> 00:03:33.590 So I want you to think of these parameters as k is a constant, 00:03:33.590 --> 00:03:36.040 so k to the some power is still a constant, 00:03:36.040 --> 00:03:40.740 but also I don't really care about polynomial changes in k. 00:03:40.740 --> 00:03:43.750 So I-- you know, we should ignore polynomial changes in k 00:03:43.750 --> 00:03:46.900 and view this constant more or less as the original k itself. 00:03:46.900 --> 00:03:51.140 So if some is-- the a plus a is around the same size as a, 00:03:51.140 --> 00:03:56.470 then further iterations also do not change the sizes very much. 00:03:56.470 --> 00:03:58.470 Ruzsa covering lemma: so this was some statement 00:03:58.470 --> 00:04:03.550 that if x plus b looks like it could be covered by copies 00:04:03.550 --> 00:04:06.580 of b, just in terms of their sizes alone, then in fact, 00:04:06.580 --> 00:04:09.760 x could be covered by a small number of translates 00:04:09.760 --> 00:04:11.560 of a slightly larger ball. 00:04:11.560 --> 00:04:14.110 But here B can be any set. 00:04:14.110 --> 00:04:16.870 We had a thing called Ruzsa modeling lemma. 00:04:16.870 --> 00:04:18.370 In particular, a consequence of it 00:04:18.370 --> 00:04:21.100 is that if a has small doubling, then there 00:04:21.100 --> 00:04:23.290 exists the prime n that's not too much 00:04:23.290 --> 00:04:26.450 bigger than the size of a, and a very large proportion-- 00:04:26.450 --> 00:04:29.440 an eighth of a subset of an eighth of a such 00:04:29.440 --> 00:04:32.080 that this subset a prime is prime 00:04:32.080 --> 00:04:34.500 and 8 isomorphic to a subset of z mod n. 00:04:34.500 --> 00:04:36.250 So even though you start with a set that's 00:04:36.250 --> 00:04:38.350 potentially very spread out, provided 00:04:38.350 --> 00:04:40.990 they have small doubling, I can pick out 00:04:40.990 --> 00:04:43.540 a pretty large piece of it and model it 00:04:43.540 --> 00:04:46.720 by something in a fairly small cyclic group. 00:04:46.720 --> 00:04:48.530 And here the modeling is 8 isomorphic, 00:04:48.530 --> 00:04:52.060 so it preserves sums up to eight term sums. 00:04:55.060 --> 00:04:56.730 We had Bogolyubovs lemma so now we're 00:04:56.730 --> 00:04:59.310 inside a small cyclic group of a large subset 00:04:59.310 --> 00:05:01.200 of a small cyclic group. 00:05:01.200 --> 00:05:05.820 Then Bogolyubovs lemma says that 2a minus 2a contains 00:05:05.820 --> 00:05:10.080 a large bore set, of large structure within the situated 00:05:10.080 --> 00:05:11.210 subset. 00:05:11.210 --> 00:05:14.270 And last time we showed that the geometry of numbers, 00:05:14.270 --> 00:05:16.470 Minkowskis second term, one can deduce 00:05:16.470 --> 00:05:22.800 that every bore set of small dimension and small width 00:05:22.800 --> 00:05:24.310 contains-- 00:05:24.310 --> 00:05:31.980 of large width contains a proper GAP that's pretty large. 00:05:31.980 --> 00:05:34.650 So putting these two together, putting the last two things 00:05:34.650 --> 00:05:39.480 together, we obtain that if you have 00:05:39.480 --> 00:05:47.690 a subset of the cyclic group and n is prime-- 00:05:47.690 --> 00:05:52.670 OK, so here in previous statement n is prime. 00:05:52.670 --> 00:05:53.630 So n is prime. 00:05:56.591 --> 00:05:57.950 And a is pretty large. 00:06:00.680 --> 00:06:10.910 Then 2a minus 2a contains a proper generalized arithmetic 00:06:10.910 --> 00:06:18.560 progression of dimension at most alpha to the minus 2 00:06:18.560 --> 00:06:28.014 and size at least 1 over 40 to the d times n. 00:06:31.250 --> 00:06:33.510 So it's just starting from the size of a. 00:06:33.510 --> 00:06:36.390 2a minus 2a contains a pretty large GAP. 00:06:36.390 --> 00:06:38.640 So we're going to put all of these ingredients 00:06:38.640 --> 00:06:40.380 together and show that you can now 00:06:40.380 --> 00:06:46.580 contain the original set, a, in a small GAP. 00:06:46.580 --> 00:06:51.130 So just from knowing that some subset of it, 2a minus 2a, 00:06:51.130 --> 00:06:55.030 so think 2a prime minus 2a prime, contains this large GAP. 00:06:55.030 --> 00:06:57.030 We're going to use it to boost it up to a cover. 00:07:05.760 --> 00:07:08.160 So now let's prove Freiman's theorem. 00:07:13.940 --> 00:07:16.024 Using the modeling lemma-- 00:07:21.440 --> 00:07:24.740 using the modeling lemma-- the corollary of the modeling 00:07:24.740 --> 00:07:32.630 lemma-- we find that since a plus a is size 00:07:32.630 --> 00:07:35.760 at most k kind of size of a, there 00:07:35.760 --> 00:07:45.510 exists some prime n at most 2k to the 16 times a. 00:07:45.510 --> 00:07:51.050 And so I'm just copying the consequence of this modeling 00:07:51.050 --> 00:07:51.900 lemma. 00:07:51.900 --> 00:07:56.190 So I find a pretty large subset of a such that a prime is prime 00:07:56.190 --> 00:08:03.170 and 8 isomorphic to a subset of z mod n. 00:08:09.550 --> 00:08:17.690 Now, applying the final corollary 00:08:17.690 --> 00:08:29.220 with alpha being the size of this a prime, which 00:08:29.220 --> 00:08:38.049 is at least the size of a over n, which is at least 1 over 16 00:08:38.049 --> 00:08:43.179 to the times k to the power 16, so all constants. 00:08:43.179 --> 00:08:48.050 We see that 2a prime minus-- 00:08:48.050 --> 00:08:49.970 so let me actually-- let me change the letters 00:08:49.970 --> 00:08:54.380 and call a prime b so I don't have to keep on writing primes. 00:08:57.450 --> 00:09:02.090 So subset of a is called b. 00:09:02.090 --> 00:09:07.575 OK, so 2b minus 2b now contains a large GAP. 00:09:12.850 --> 00:09:18.800 And the GAP has dimension d bounded. 00:09:18.800 --> 00:09:22.370 So the dimension is bounded by alpha to the minus 2. 00:09:22.370 --> 00:09:23.755 So it's some constant. 00:09:28.880 --> 00:09:32.490 And the size is pretty large. 00:09:32.490 --> 00:09:39.775 So size is at least 1 over 40 d. 00:09:39.775 --> 00:09:41.150 If you only care about constants, 00:09:41.150 --> 00:09:44.900 just remember that everything that depends on k or d 00:09:44.900 --> 00:09:45.760 is a constant. 00:09:49.580 --> 00:09:50.080 OK. 00:09:53.920 --> 00:10:00.370 Because b is Freiman's 8 isomorphic, 00:10:00.370 --> 00:10:06.340 b is Freiman 8 isomorphic to-- 00:10:11.080 --> 00:10:13.450 ah, sorry. 00:10:13.450 --> 00:10:23.800 b is-- a prime is a subset of a and b is the subset of z mod-- 00:10:23.800 --> 00:10:28.540 b is a subset of z mod n that a prime is 8 isomorphic, too. 00:10:28.540 --> 00:10:37.330 So since b is 8 isomorphic to a prime, every GAP in b-- 00:10:40.330 --> 00:10:45.360 so if you think about what 8 isomorphism preserves, 00:10:45.360 --> 00:10:53.880 you find that if you look at 2b minus 2b, it must be 2 prime-- 00:10:53.880 --> 00:11:00.760 2 isomorphic to 2a prime minus 2a prime. 00:11:00.760 --> 00:11:02.730 So the point of prime and isomorphism 00:11:02.730 --> 00:11:07.565 is that we just want to preserve enough additive structure. 00:11:07.565 --> 00:11:09.940 Well, we're doing to preserve all the additive structure, 00:11:09.940 --> 00:11:11.357 but just enough additive structure 00:11:11.357 --> 00:11:13.330 to do what we need to do. 00:11:13.330 --> 00:11:16.150 And being able to preserve an arithmetic progression, 00:11:16.150 --> 00:11:18.820 or in general or generalized arithmetic progression, 00:11:18.820 --> 00:11:23.400 requires you to preserve Freiman 2 isomorphism. 00:11:23.400 --> 00:11:25.280 And that's where the a comes in. 00:11:25.280 --> 00:11:28.180 So I want to analyze 2b minus 2b and I 00:11:28.180 --> 00:11:30.400 want that to preserve 2 isomorphisms. 00:11:30.400 --> 00:11:36.000 So initially I want b to preserve 8 isomorphisms. 00:11:36.000 --> 00:11:40.690 So 2b minus 2b is Freiman isomorphic to 2a prime minus 2a 00:11:40.690 --> 00:11:42.070 prime. 00:11:42.070 --> 00:11:50.690 So the GAP, which we found earlier in 2B minus 2B 00:11:50.690 --> 00:11:58.830 is mapped via this Freiman isomorphism 00:11:58.830 --> 00:12:04.880 to a proper GAP, which we'll call q, 00:12:04.880 --> 00:12:11.580 now setting aside 2a minus 2a and preserving 00:12:11.580 --> 00:12:15.456 the same dimension and size. 00:12:15.456 --> 00:12:19.200 So Freiman isomorphisms are good for preserving 00:12:19.200 --> 00:12:21.900 these partial additive structures like GAPs. 00:12:21.900 --> 00:12:22.610 Yes? 00:12:22.610 --> 00:12:25.060 AUDIENCE: So are we using this smaller structure to be 00:12:25.060 --> 00:12:26.040 [INAUDIBLE]? 00:12:33.357 --> 00:12:34.190 YUFEI ZHAO: Correct. 00:12:34.190 --> 00:12:37.430 So question is, we're using-- 00:12:37.430 --> 00:12:41.630 so because we have to pass the 2b minus 2b, 00:12:41.630 --> 00:12:44.600 we want 2b minus 2b to be prime and isomorphic to 2a 00:12:44.600 --> 00:12:45.710 prime minus 2a prime. 00:12:48.240 --> 00:12:52.730 So that's why in the proof I want b 00:12:52.730 --> 00:12:55.500 to be 8 isomorphic to a prime. 00:12:55.500 --> 00:12:58.800 So you see, so I'm skipping details of this step. 00:12:58.800 --> 00:13:02.330 But if you read the definition of Freiman's s isomorphism, 00:13:02.330 --> 00:13:04.111 you see that this implication holds. 00:13:04.111 --> 00:13:06.465 AUDIENCE: [INAUDIBLE] 00:13:06.465 --> 00:13:07.090 YUFEI ZHAO: No. 00:13:07.090 --> 00:13:10.367 2 isomorphism is a weaker condition. 00:13:13.110 --> 00:13:17.170 2 isomorphism just means that you are preserving two y sums. 00:13:21.030 --> 00:13:25.870 So think about the definition of Freiman 2 isomorphisms. 00:13:25.870 --> 00:13:30.160 In particular, if two sets are Freiman 2 isomorphic 00:13:30.160 --> 00:13:33.160 and you have a arithmetic progression in one, 00:13:33.160 --> 00:13:35.620 then that arithmetic progression is also an arithmetic 00:13:35.620 --> 00:13:38.800 progression in the other. 00:13:38.800 --> 00:13:41.310 So it's just enough additive structure 00:13:41.310 --> 00:13:43.860 to preserve things like arithmetic progressions 00:13:43.860 --> 00:13:48.456 and generalized arithmetic progressions. 00:13:48.456 --> 00:13:49.360 OK. 00:13:49.360 --> 00:13:56.020 So we found this large GAP in 2a minus 2a. 00:13:56.020 --> 00:13:58.180 So this is very good. 00:13:58.180 --> 00:14:01.540 So we wanted to contain a in the GAP. 00:14:01.540 --> 00:14:03.800 Seems that by now we're doing something slightly 00:14:03.800 --> 00:14:04.900 in the opposite direction. 00:14:04.900 --> 00:14:10.917 We find a large GAP within 2a minus 2a. 00:14:10.917 --> 00:14:12.500 But something we've seen before, we're 00:14:12.500 --> 00:14:16.100 going to use this to boost ourselves to a covering of a 00:14:16.100 --> 00:14:18.750 via the Ruzsa covering lemma. 00:14:18.750 --> 00:14:21.540 So once you find this large structure, 00:14:21.540 --> 00:14:27.800 you can now try to take translates of it to cover a. 00:14:27.800 --> 00:14:30.170 And this is-- if there's any takeaway from the spirit 00:14:30.170 --> 00:14:31.970 of this proof is this idea. 00:14:31.970 --> 00:14:34.160 Even though I want to cover the whole set, it's OK. 00:14:34.160 --> 00:14:37.220 I just find a large structure within it 00:14:37.220 --> 00:14:39.140 and then I use translaters to cover. 00:14:41.770 --> 00:14:43.450 How do we do this? 00:14:43.450 --> 00:14:50.350 So since q is containing 2a minus 2a, 00:14:50.350 --> 00:14:53.650 we find that q plus a is containing 3a minus 2a. 00:15:00.120 --> 00:15:07.110 Therefore, by Plünnecke-Ruzsa-- by Plünnecke-Ruzsa inequality, 00:15:07.110 --> 00:15:13.790 the size of cube plus a is at most the size of 3a minus 2a, 00:15:13.790 --> 00:15:17.720 which is, at most, k to the fifth power times the size 00:15:17.720 --> 00:15:18.220 of a. 00:15:23.120 --> 00:15:27.060 And I claim that this final quantity is also not 00:15:27.060 --> 00:15:34.120 so different from the size of cube, because-- 00:15:34.120 --> 00:15:35.590 so all of these-- 00:15:35.590 --> 00:15:37.672 I mean, the point here is, we are 00:15:37.672 --> 00:15:39.130 doing all of these transformations, 00:15:39.130 --> 00:15:41.830 passing down to subsets, putting something bigger, putting-- 00:15:41.830 --> 00:15:44.260 getting to something smaller, but each time we only 00:15:44.260 --> 00:15:47.910 loose something that is polynomial in k. 00:15:47.910 --> 00:15:49.700 We're not losing much more. 00:15:49.700 --> 00:15:52.450 I am also only losing a constant factor. 00:15:52.450 --> 00:15:56.110 There is sometimes a bit more than polynomial, 00:15:56.110 --> 00:16:01.280 but any case, we're losing only a constant factor at each step. 00:16:01.280 --> 00:16:12.110 So in particular, since n upper bounds the size of a prime 00:16:12.110 --> 00:16:17.610 is here where we ended up embedding into z modern, 00:16:17.610 --> 00:16:23.270 n is larger than a prime, which is at least a constant fraction 00:16:23.270 --> 00:16:31.390 of a and the size of q is at least 1 00:16:31.390 --> 00:16:38.210 over 40 raised d times n. 00:16:38.210 --> 00:16:43.870 So we find that this bound-- 00:16:43.870 --> 00:16:46.980 upper bound earlier on q plus a. 00:16:46.980 --> 00:16:56.360 We can write it in terms of size of cube 00:16:56.360 --> 00:17:01.760 where k prime is-- you put all of these numbers together. 00:17:01.760 --> 00:17:05.180 What it is specifically doesn't matter so much, 00:17:05.180 --> 00:17:06.589 other than that it is a constant. 00:17:11.829 --> 00:17:14.359 d is polynomial in k. 00:17:14.359 --> 00:17:18.050 So what we have here is something that is exponential, 00:17:18.050 --> 00:17:19.069 polynomial of k. 00:17:27.790 --> 00:17:31.110 OK, so now we're in a position to apply the Ruzsa covering 00:17:31.110 --> 00:17:32.400 lemma. 00:17:32.400 --> 00:17:35.440 So look at that statement up there. 00:17:35.440 --> 00:17:37.210 So what is the saying, that a plus q 00:17:37.210 --> 00:17:41.310 looks like it could be covered by q, just in terms of size. 00:17:41.310 --> 00:17:45.450 So I should expect to cover a by a small number of translates 00:17:45.450 --> 00:17:46.590 of q minus q. 00:17:51.430 --> 00:17:57.690 So by covering lemma, a is containing some x plus q 00:17:57.690 --> 00:18:05.630 minus q for some x in a where the size of x 00:18:05.630 --> 00:18:07.640 is at most k prime. 00:18:11.350 --> 00:18:14.920 I claim-- so we've covered a by something. 00:18:14.920 --> 00:18:18.740 q is a GAP. 00:18:18.740 --> 00:18:22.690 x is a bounded size set, and I claim 00:18:22.690 --> 00:18:27.950 that this is the type of object that we're happy to have. 00:18:27.950 --> 00:18:31.980 Just to spell out some details, first 00:18:31.980 --> 00:18:45.480 note that x is contained in a GAP dimension x or x minus 1 00:18:45.480 --> 00:18:48.217 with length 2 in each direction. 00:18:54.560 --> 00:18:56.820 So add a new direction for every element of x. 00:18:56.820 --> 00:19:00.450 It's wasteful, but everything's constant. 00:19:00.450 --> 00:19:07.230 And recall that the dimension of Q as a GAP is d. 00:19:07.230 --> 00:19:22.980 So x plus q minus q is contained in a GAP of dimension. 00:19:22.980 --> 00:19:24.900 OK, so what's the dimension? 00:19:24.900 --> 00:19:28.890 So when I do q minus q, it's like taking a box 00:19:28.890 --> 00:19:32.490 and doubling its dimension-- doubling its lengths. 00:19:32.490 --> 00:19:34.870 I'm not changing the number of dimensions. 00:19:34.870 --> 00:19:37.530 So the dimension of q minus q is still d. 00:19:37.530 --> 00:19:41.250 The dimension of x is, at most, the size of x. 00:19:44.220 --> 00:19:46.850 All of these things are constants. 00:19:46.850 --> 00:19:47.755 So we're happy. 00:19:47.755 --> 00:19:52.210 But to spell it out, the constant here is-- 00:19:52.210 --> 00:19:55.800 well, k prime is what we wrote up there. 00:19:59.800 --> 00:20:04.040 So this is a constant. 00:20:04.040 --> 00:20:11.570 And the size-- so what is the size of the GAP 00:20:11.570 --> 00:20:13.940 that contains this guy here? 00:20:13.940 --> 00:20:19.730 So I'm expanding x to a GAP by adding a new direction 00:20:19.730 --> 00:20:22.340 for every element of x. 00:20:22.340 --> 00:20:25.250 And I might expand that size a little bit. 00:20:25.250 --> 00:20:29.930 But the size of this GAP that contains x is no more than 2 00:20:29.930 --> 00:20:33.100 to the power x-- 00:20:33.100 --> 00:20:37.220 2 raised to the size of x. 00:20:37.220 --> 00:20:42.860 What is the size of GAP q minus q? 00:20:42.860 --> 00:20:45.740 So q is the GAP of dimension d. 00:20:45.740 --> 00:20:48.050 And we know that a GAP of dimension d 00:20:48.050 --> 00:20:52.710 has doubling constant and those 2 to the d-- 00:20:52.710 --> 00:20:56.960 2 to the d times the size of q. 00:20:56.960 --> 00:20:58.060 OK. 00:20:58.060 --> 00:21:03.580 And because q is contained in qa minus 2a, 00:21:03.580 --> 00:21:10.340 we find that q is contained in 2a minus 2a. 00:21:10.340 --> 00:21:12.830 And 2 to the x, well, I know what the site of x 00:21:12.830 --> 00:21:22.800 is bounded by, it's k prime plus the size of x is-- 00:21:22.800 --> 00:21:24.390 size of x is, at most, k prime. 00:21:24.390 --> 00:21:27.890 And then I have 2 to the d over here, 00:21:27.890 --> 00:21:34.150 so 2a minus 2a by Plünnecke-Ruzsa is at most k 00:21:34.150 --> 00:21:36.490 to the 4 times the size of a. 00:21:36.490 --> 00:21:39.820 OK, you put everything together, we find that this bound here 00:21:39.820 --> 00:21:44.790 is doubly exponential in the-- it's a polynomial in k. 00:21:47.920 --> 00:21:50.490 And that's it. 00:21:50.490 --> 00:21:52.240 This proves Freiman's theory. 00:21:58.792 --> 00:22:03.785 Now, to recap-- we went through several steps. 00:22:03.785 --> 00:22:07.050 So first, using the modeling lemma, 00:22:07.050 --> 00:22:09.990 we know that if a set a has small doubling, 00:22:09.990 --> 00:22:14.940 then we can pass a large part of a to a relatively 00:22:14.940 --> 00:22:17.290 small cyclic group. 00:22:17.290 --> 00:22:20.710 Going to work inside our cyclic group. 00:22:20.710 --> 00:22:28.220 Using Bogolyubovs lemma and its geometry of numbers corollary, 00:22:28.220 --> 00:22:30.890 we find that inside the cyclic group, 00:22:30.890 --> 00:22:33.950 the corresponding set, which we called b, 00:22:33.950 --> 00:22:37.330 is such that 2b minus 2b contains a large GAP. 00:22:40.390 --> 00:22:43.450 We pass that GAP back to the original set 00:22:43.450 --> 00:22:49.290 a because we are preserving 8 isomorphisms Freiman 8 00:22:49.290 --> 00:22:51.260 isomorphisms in Ruzsa modeling lemma 00:22:51.260 --> 00:22:53.600 so we can pass to the original set a 00:22:53.600 --> 00:22:58.770 and find the large GAP in 2a minus 2a. 00:22:58.770 --> 00:23:02.880 Once we find this large GAP 2a minus 2a, 00:23:02.880 --> 00:23:06.510 then we're going to use the Ruzsa covering 00:23:06.510 --> 00:23:15.390 lemma to contain a inside a small number of translates 00:23:15.390 --> 00:23:16.560 of this GAP. 00:23:20.060 --> 00:23:20.750 OK. 00:23:20.750 --> 00:23:23.450 You put all of these things together and the appropriate 00:23:23.450 --> 00:23:27.310 bounds coming from Plünnecke-Ruzsa inequalities 00:23:27.310 --> 00:23:30.218 and you get a final theorem. 00:23:30.218 --> 00:23:32.010 And this is the proof of Freiman's theorem. 00:23:34.735 --> 00:23:36.185 AUDIENCE: [INAUDIBLE] 00:23:36.185 --> 00:23:36.810 YUFEI ZHAO: OK. 00:23:36.810 --> 00:23:40.420 The question is how do you make it proper? 00:23:40.420 --> 00:23:44.320 Up until the step with q, it is still proper. 00:23:44.320 --> 00:23:48.790 So the very last step over here it is-- 00:23:48.790 --> 00:23:51.610 you might have destroyed properness. 00:23:51.610 --> 00:23:54.410 So this proof here doesn't give you properness. 00:23:54.410 --> 00:23:58.000 So I mentioned at the beginning that in Freiman's theorem, 00:23:58.000 --> 00:24:02.170 you can obtain properness of the additional arguments. 00:24:02.170 --> 00:24:03.660 So that I'm not going to show. 00:24:03.660 --> 00:24:08.125 There's some more work which is related to geometry of numbers. 00:24:10.940 --> 00:24:13.825 So for example, you can look up in the textbook by Tao and Vu, 00:24:13.825 --> 00:24:21.340 and see how to get from a GAP to contain it in a proper GAP, 00:24:21.340 --> 00:24:25.780 without losing too much in terms of size. 00:24:25.780 --> 00:24:28.532 So think about it this way-- 00:24:28.532 --> 00:24:30.490 when do you have something which is not proper? 00:24:30.490 --> 00:24:33.160 When you have-- and if some linear dependence, 00:24:33.160 --> 00:24:35.680 you have some integer linear dependence. 00:24:35.680 --> 00:24:40.060 And in that case, you kind of lost a dimension. 00:24:40.060 --> 00:24:42.940 When you have improperness, you actually go down a dimension. 00:24:42.940 --> 00:24:46.270 But then you need to salvage the size, 00:24:46.270 --> 00:24:48.830 make sure that the size doesn't blow up too much. 00:24:48.830 --> 00:24:50.960 And so there are some arguments to be done there. 00:24:50.960 --> 00:24:52.810 And we're not going to do it here. 00:24:52.810 --> 00:24:54.960 AUDIENCE: Well, I guess my [INAUDIBLE] 00:24:54.960 --> 00:24:58.880 do they change [INAUDIBLE] within the proof, 00:24:58.880 --> 00:25:01.510 like say [INAUDIBLE] q or whatever? 00:25:01.510 --> 00:25:05.060 Or do they use the same proof, but later on, 00:25:05.060 --> 00:25:08.725 say that [INAUDIBLE]? 00:25:08.725 --> 00:25:09.600 YUFEI ZHAO: OK, good. 00:25:09.600 --> 00:25:11.340 Yeah, so the question is, to get properness, 00:25:11.340 --> 00:25:13.882 do I have to modify the proof, or can I use Freiman's theorem 00:25:13.882 --> 00:25:15.350 as witness of a black box. 00:25:15.350 --> 00:25:17.020 So my understanding is that I can 00:25:17.020 --> 00:25:20.422 use the statement as a black box and obtain properness. 00:25:20.422 --> 00:25:21.880 But if you want to get good bounds, 00:25:21.880 --> 00:25:24.250 maybe you have to go into the proof, although that, 00:25:24.250 --> 00:25:24.820 I'm not sure. 00:25:27.450 --> 00:25:31.500 OK, any more questions? 00:25:31.500 --> 00:25:34.670 So this took a while. 00:25:34.670 --> 00:25:38.620 This was the most involved proof we've done in this course 00:25:38.620 --> 00:25:40.630 so far, in proving Freiman's theorem. 00:25:40.630 --> 00:25:43.500 We had to develop a large number of tools. 00:25:43.500 --> 00:25:44.930 And we came up-- 00:25:44.930 --> 00:25:47.140 so we eventually arrived at-- 00:25:47.140 --> 00:25:48.260 it's a beautiful theorem. 00:25:48.260 --> 00:25:51.190 So this is a fantastic result that 00:25:51.190 --> 00:25:55.030 gives you an inverse structure, something-- 00:25:55.030 --> 00:25:57.520 we know that GAP's have small doubling. 00:25:57.520 --> 00:26:00.170 And conversely, if something has a small doubling, 00:26:00.170 --> 00:26:03.650 it has to, in some sense, look like a GAP. 00:26:03.650 --> 00:26:05.590 So you see that the proof is quite involved 00:26:05.590 --> 00:26:08.780 and has a lot of beautiful ideas. 00:26:08.780 --> 00:26:10.690 In the remainder of today's lecture, 00:26:10.690 --> 00:26:15.250 I want to present some remarks on additional extensions 00:26:15.250 --> 00:26:17.710 and generalizations of Freiman's theorem. 00:26:17.710 --> 00:26:20.660 And while we're not going to do any proofs, 00:26:20.660 --> 00:26:23.230 there's a lot of deep and beautiful mathematics 00:26:23.230 --> 00:26:26.230 that are involved in the subject. 00:26:26.230 --> 00:26:31.150 So I want to take you on a tour through some more things 00:26:31.150 --> 00:26:34.480 that we can talk about when it comes to Freiman's theorem. 00:26:34.480 --> 00:26:38.350 But first, let me mention a few things 00:26:38.350 --> 00:26:42.040 that I mentioned very quickly when we first 00:26:42.040 --> 00:26:43.990 introduced Freiman's theorem, namely 00:26:43.990 --> 00:26:45.310 some remarks on the bounds. 00:26:52.120 --> 00:26:55.890 So the proof that we just saw gives you 00:26:55.890 --> 00:27:00.550 a bound, which is basically exponential in the dimension 00:27:00.550 --> 00:27:04.770 and doubly exponential for the size blow-up. 00:27:04.770 --> 00:27:08.530 They're all constants, so if you only care about constants, 00:27:08.530 --> 00:27:10.540 then this is just fine. 00:27:10.540 --> 00:27:13.790 But you may ask, are we losing too much here? 00:27:13.790 --> 00:27:18.120 What is the right type of dependence? 00:27:18.120 --> 00:27:20.610 So what is the right type of dependence? 00:27:20.610 --> 00:27:23.030 So we saw an example. 00:27:23.030 --> 00:27:28.020 So we saw an example where if you start with A being 00:27:28.020 --> 00:27:35.930 a highly dissociated set, where there is basically 00:27:35.930 --> 00:27:40.370 no additive structure within A, then you do need-- 00:27:40.370 --> 00:27:49.600 so this example shows that you cannot do better than 00:27:49.600 --> 00:27:50.900 polynomial-- 00:27:50.900 --> 00:27:58.080 well, actually, than linear in K, in the dimension, 00:27:58.080 --> 00:28:03.240 and exponential in the size blow-up. 00:28:03.240 --> 00:28:06.030 So in particular, you do need to blow up 00:28:06.030 --> 00:28:11.690 the size by some exponential quantity in K. 00:28:11.690 --> 00:28:16.450 So here, K is roughly the size of A over 2 in this example. 00:28:16.450 --> 00:28:19.130 And you can create modifications of the example 00:28:19.130 --> 00:28:22.020 to keep K constant and A getting larger. 00:28:22.020 --> 00:28:25.160 But the point is that you cannot do better than this type 00:28:25.160 --> 00:28:27.740 of dependence simply from that example. 00:28:27.740 --> 00:28:31.100 And it's conjecture that that is the truth. 00:28:31.100 --> 00:28:33.590 We're almost there in proving this conjecture, 00:28:33.590 --> 00:28:36.020 but not quite, although the proof that we just gave 00:28:36.020 --> 00:28:41.400 is somewhat far, because you lose an exponent in each bound. 00:28:41.400 --> 00:28:45.210 There is a refinement of the final step in the argument, 00:28:45.210 --> 00:28:46.680 so let me comment on that. 00:28:46.680 --> 00:28:53.750 So we can refine the final step or the final steps 00:28:53.750 --> 00:28:59.045 in the proof to get polynomial bounds. 00:29:04.000 --> 00:29:10.690 And to get a polynomial bounds, which 00:29:10.690 --> 00:29:13.570 is much more in the right ballpark 00:29:13.570 --> 00:29:15.710 compared to what we got. 00:29:15.710 --> 00:29:18.820 And the idea is basically over here, 00:29:18.820 --> 00:29:20.380 we used the Ruzsa covering lemma. 00:29:20.380 --> 00:29:23.500 So we started with that Q up there. 00:29:23.500 --> 00:29:30.660 So up until this point, you should think of this step 00:29:30.660 --> 00:29:34.390 as everything coming from Bogolyubov and its corollary. 00:29:34.390 --> 00:29:36.650 So that stays the same. 00:29:36.650 --> 00:29:39.830 And now the question is starting with our Q, what would you use? 00:29:39.830 --> 00:29:45.340 How would you use this Q to try to cover it? 00:29:45.340 --> 00:29:48.673 Well, what we do, we apply Ruzsa covering lemma. 00:29:48.673 --> 00:29:50.840 Remember how the proof of Ruzsa covering lemma goes. 00:29:50.840 --> 00:29:54.920 You take a maximal set of translates, 00:29:54.920 --> 00:29:56.260 disjoint translates. 00:29:56.260 --> 00:29:58.010 And if you blow everything up a factor 2, 00:29:58.010 --> 00:30:00.500 then you've got a cover. 00:30:00.500 --> 00:30:03.030 But it turns out to be somewhat wasteful. 00:30:03.030 --> 00:30:06.860 And you see, there was a lot of waste in going from x to 2 00:30:06.860 --> 00:30:08.760 to the x. 00:30:08.760 --> 00:30:12.550 So you could do that step more slowly. 00:30:12.550 --> 00:30:21.100 So starting with Q, cover now some, not all 00:30:21.100 --> 00:30:34.500 of A. So cover parts of A by translates of 2 minus Q, say. 00:30:34.500 --> 00:30:36.390 So we do Ruzsa covering lemma, you 00:30:36.390 --> 00:30:40.110 don't cover the whole thing, but nibble away, cover 00:30:40.110 --> 00:30:44.870 a little bit, and then look at the thing that you get, 00:30:44.870 --> 00:30:51.230 which is that Q will become some new thing, let's say Q1. 00:30:51.230 --> 00:31:00.530 And now cover more by Q1 minus Q1. 00:31:00.530 --> 00:31:06.670 So apparently, if you do the covering step more slowly, 00:31:06.670 --> 00:31:08.530 you can obtain better bounds. 00:31:08.530 --> 00:31:14.320 And that's enough to save you this exponent, 00:31:14.320 --> 00:31:19.230 to go down to polynomial-type bounds for Freiman's theorem. 00:31:19.230 --> 00:31:21.830 So I'm not giving details, but this is roughly the idea. 00:31:21.830 --> 00:31:26.250 So you can modify the final step to obtain this bound. 00:31:26.250 --> 00:31:35.410 The best bound so far is due to Tom Sanders, who 00:31:35.410 --> 00:31:40.300 proved Freiman's theorem for bounds 00:31:40.300 --> 00:31:46.540 on dimension that's like K times poly log K, 00:31:46.540 --> 00:31:54.310 and the size blowup to be E to the K times poly log K. 00:31:54.310 --> 00:31:57.310 So in other words, other than this polylogarithmic factor, 00:31:57.310 --> 00:32:00.740 it's basically the right answer. 00:32:00.740 --> 00:32:04.220 And so this proof is much more sophisticated. 00:32:04.220 --> 00:32:07.390 So it goes much more in depth into analyzing 00:32:07.390 --> 00:32:09.530 the structure of set addition. 00:32:09.530 --> 00:32:12.250 So Sanders has a very nice survey article 00:32:12.250 --> 00:32:14.500 called "The structure of Set Addition" 00:32:14.500 --> 00:32:19.090 that analyzes some of the modern techniques that 00:32:19.090 --> 00:32:21.180 are used to prove these types of results. 00:32:28.480 --> 00:32:30.220 There is one more issue, which I want 00:32:30.220 --> 00:32:33.360 to discuss at length in the second half of this lecture, 00:32:33.360 --> 00:32:36.940 which is that you might be very unhappy 00:32:36.940 --> 00:32:41.770 with this exponential blowup, because if you think about what 00:32:41.770 --> 00:32:43.720 happens in these examples-- 00:32:43.720 --> 00:32:46.440 I mean, not the examples, but if you think about what happens, 00:32:46.440 --> 00:32:49.390 like the spirit of what we're trying to say, 00:32:49.390 --> 00:32:53.390 Freiman's theorem is some kind of an inverse theorem. 00:32:53.390 --> 00:32:57.410 And to go forward, you're trying to say 00:32:57.410 --> 00:33:05.060 that if you have a GAP of dimension d, then 00:33:05.060 --> 00:33:11.000 the size blowup is like 2 to the d. 00:33:11.000 --> 00:33:22.130 So we want to say some structure applies small doubling, 00:33:22.130 --> 00:33:25.200 and Freiman's theorem tells the reverse, that you 00:33:25.200 --> 00:33:33.030 have small doubling, then you obtain this structure. 00:33:33.030 --> 00:33:39.420 And seems like you are losing. 00:33:39.420 --> 00:33:44.340 Getting from here to here, there is a polynomial type of loss, 00:33:44.340 --> 00:33:46.290 whereas going from here to here, it 00:33:46.290 --> 00:33:49.950 seems that we're incurring some exponential type of loss. 00:33:49.950 --> 00:33:52.740 And it'll be nice to have some kind of inverse theorem 00:33:52.740 --> 00:33:57.180 that also preserves these relationships qualitatively. 00:33:57.180 --> 00:33:59.070 So that may not make sense in this moment, 00:33:59.070 --> 00:34:00.903 but we'll get back to it later this lecture. 00:34:04.250 --> 00:34:05.660 Point is, there's more, much more 00:34:05.660 --> 00:34:07.520 to be said about the bounds here, 00:34:07.520 --> 00:34:09.530 even though right now it looks as if they're 00:34:09.530 --> 00:34:12.860 very close to each other. 00:34:12.860 --> 00:34:15.980 One more thing that I want to expand on 00:34:15.980 --> 00:34:19.300 is, we've stated and proved Freiman's theorem 00:34:19.300 --> 00:34:21.409 in the integers. 00:34:21.409 --> 00:34:26.850 And you might ask, what about in other groups? 00:34:26.850 --> 00:34:31.620 We also proved Freiman's theorem in F2 to the m, or more 00:34:31.620 --> 00:34:33.989 generally, groups of bounded exponent 00:34:33.989 --> 00:34:39.270 or bounded portion, so abelian groups of bounded exponent. 00:34:39.270 --> 00:34:46.600 For general abelian groups, so Freiman's theorem 00:34:46.600 --> 00:34:54.900 in general abelian groups, you might ask what happens here? 00:34:54.900 --> 00:34:58.230 And in some sense what is even the statement of the theorem? 00:34:58.230 --> 00:35:01.120 So we want something which combines, 00:35:01.120 --> 00:35:04.280 somehow, two different types of behavior. 00:35:04.280 --> 00:35:07.670 On one hand, you have z, which is what we just did. 00:35:07.670 --> 00:35:10.810 And here the model structures are GAP's. 00:35:10.810 --> 00:35:14.330 And on the other hand, we have, which we also proved, 00:35:14.330 --> 00:35:17.570 things like F2 to the m, where the model structures are 00:35:17.570 --> 00:35:18.200 subgroups. 00:35:22.520 --> 00:35:24.270 And there's a sense in which these are not 00:35:24.270 --> 00:35:26.400 the GAP's and subgroups. 00:35:26.400 --> 00:35:27.900 They have some similar properties, 00:35:27.900 --> 00:35:31.350 but they're not really like each other. 00:35:31.350 --> 00:35:33.500 So now if I give you a general group, which 00:35:33.500 --> 00:35:36.740 might be some combination of infinite torsion 00:35:36.740 --> 00:35:39.590 or very large torsion elements versus very small torsion 00:35:39.590 --> 00:35:42.410 elements-- so for example, take a Cartesian product 00:35:42.410 --> 00:35:43.820 of these groups. 00:35:43.820 --> 00:35:45.900 Is there a Freiman's theorem? 00:35:45.900 --> 00:35:48.780 And what does such a theorem look like? 00:35:48.780 --> 00:35:51.340 What are the structures? 00:35:51.340 --> 00:35:54.080 What are the subsets of bounded doubling? 00:35:58.040 --> 00:36:01.590 So that's kind of the thing we want to think about. 00:36:01.590 --> 00:36:05.010 So it turns out for Freiman's theorem in general abelian 00:36:05.010 --> 00:36:06.330 groups-- 00:36:06.330 --> 00:36:07.570 so there is a theorem. 00:36:07.570 --> 00:36:10.950 So this theorem was proved by Green and Ruzsa. 00:36:14.880 --> 00:36:17.730 So following a very similar type of proof framework, 00:36:17.730 --> 00:36:21.330 although the individual steps, in particular 00:36:21.330 --> 00:36:24.840 the modeling lemma needs to be modified. 00:36:24.840 --> 00:36:27.180 And let me tell you what the statement is. 00:36:27.180 --> 00:36:32.200 So the common generalization of GAP's and subgroups is 00:36:32.200 --> 00:36:34.740 something called a "co-set progression." 00:36:41.330 --> 00:36:46.790 So a co-set progression is a subset 00:36:46.790 --> 00:36:54.070 which is a direct sum of the form P plus H, 00:36:54.070 --> 00:36:57.180 where P is a proper GAP. 00:36:59.845 --> 00:37:03.990 So the definition of GAP works just fine in every abelian 00:37:03.990 --> 00:37:04.490 group. 00:37:04.490 --> 00:37:08.390 You start with the initial point, a few directions, 00:37:08.390 --> 00:37:13.590 and you look at a grid expansion of those directions. 00:37:13.590 --> 00:37:17.120 P is a proper GAP, and H is a subgroup. 00:37:20.300 --> 00:37:26.100 And here, the direct sum refers to the fact that every-- 00:37:26.100 --> 00:37:34.330 so if P plus H equals to P prime plus H prime for some P and P 00:37:34.330 --> 00:37:39.310 prime in the set P, and H and H prime in the set H, 00:37:39.310 --> 00:37:43.780 then P equals to P prime and H equals to H prime. 00:37:43.780 --> 00:37:46.150 So every element in here is written 00:37:46.150 --> 00:37:49.872 in a unique way as some P plus some H. So 00:37:49.872 --> 00:37:51.330 that's what I mean by "direct sum." 00:37:54.400 --> 00:37:58.930 For such an object, so such a co-set progression, 00:37:58.930 --> 00:38:09.640 I call its dimension to be the dimension of the GAP, P. 00:38:09.640 --> 00:38:12.820 And its size in this case, actually, 00:38:12.820 --> 00:38:16.930 is just the size of the set, which is also the size of P 00:38:16.930 --> 00:38:23.620 times the size of H. So the theorem 00:38:23.620 --> 00:38:36.960 is that if A is a subset of an arbitrary abelian group 00:38:36.960 --> 00:38:49.650 and it has bounded doubling, then A is contained 00:38:49.650 --> 00:39:03.930 in a co-set progression of bounded dimension and size, 00:39:03.930 --> 00:39:10.596 bounded blowup of the size of A. 00:39:10.596 --> 00:39:14.400 And here, these constants D and K are universal. 00:39:14.400 --> 00:39:17.360 They do not depend on the group. 00:39:17.360 --> 00:39:19.050 So there are some specific numbers, 00:39:19.050 --> 00:39:20.480 functions you can write down. 00:39:20.480 --> 00:39:24.040 They do not depend on the group. 00:39:24.040 --> 00:39:29.600 So this theorem gives you the characterization 00:39:29.600 --> 00:39:32.668 of subsets in general abelian groups 00:39:32.668 --> 00:39:33.710 that have small doubling. 00:39:36.610 --> 00:39:38.056 Any questions? 00:39:38.056 --> 00:39:39.808 Yes? 00:39:39.808 --> 00:39:40.690 AUDIENCE: [INAUDIBLE] 00:39:40.690 --> 00:39:41.510 YUFEI ZHAO: That's a good question. 00:39:41.510 --> 00:39:43.177 So I think you could go into their paper 00:39:43.177 --> 00:39:48.050 and see that you can get polynomial type bounds. 00:39:48.050 --> 00:39:51.920 And I think Sander's results also 00:39:51.920 --> 00:39:57.020 work for this type of setting to give you these type of bounds. 00:39:57.020 --> 00:39:59.960 But I-- yes, so you should look into Sanders' paper, 00:39:59.960 --> 00:40:00.830 and he will explain. 00:40:00.830 --> 00:40:04.200 I think in Sanders' paper he walks in general abelian 00:40:04.200 --> 00:40:04.700 groups. 00:40:07.960 --> 00:40:10.640 The next question I want to address is-- 00:40:10.640 --> 00:40:15.520 well, what do you think is the next question? 00:40:15.520 --> 00:40:20.370 Non-abelian groups, so Freiman's theorem in non-abelian groups, 00:40:20.370 --> 00:40:25.250 or rather the Freiman problem in non-abelian groups. 00:40:32.500 --> 00:40:35.710 So here's a basic question-- if I give you 00:40:35.710 --> 00:40:41.390 a non-abelian group, what subsets have bounded doubling? 00:40:41.390 --> 00:40:43.730 Of course, the examples from abelian groups 00:40:43.730 --> 00:40:47.330 also work in non-abelian groups, where you have subgroups, 00:40:47.330 --> 00:40:50.930 you have generalized arithmetical progressions. 00:40:50.930 --> 00:40:54.290 But are there genuinely new examples 00:40:54.290 --> 00:40:57.980 of sets in non-abelian groups that have bounded doubling? 00:41:01.160 --> 00:41:03.370 So think about that, and let's take a quick break. 00:41:07.790 --> 00:41:11.420 Can you think of examples in non-abelian groups 00:41:11.420 --> 00:41:14.518 that have small doubling, that do not come from the examples 00:41:14.518 --> 00:41:15.560 that we have seen before? 00:41:22.470 --> 00:41:24.090 So let me show you one construction. 00:41:24.090 --> 00:41:26.690 And this is that important construction 00:41:26.690 --> 00:41:27.740 for non-abelian groups. 00:41:34.662 --> 00:41:35.370 So it has a name. 00:41:35.370 --> 00:41:43.690 It's called a discrete Heisenberg group, 00:41:43.690 --> 00:41:51.820 which is the matrix group consisting of matrices that 00:41:51.820 --> 00:41:54.130 look like what I've written. 00:41:54.130 --> 00:41:56.740 So you have integer entries above the diagonal, 00:41:56.740 --> 00:42:01.090 1 on the diagonal, and 0 below the diagonal. 00:42:01.090 --> 00:42:03.030 So let's do some elementary matrix 00:42:03.030 --> 00:42:08.680 multiplication to see how group multiplication in this group 00:42:08.680 --> 00:42:10.190 works. 00:42:10.190 --> 00:42:17.910 So if I have two such matrices, I multiply them together. 00:42:17.910 --> 00:42:20.475 And then you see that the diagonal 00:42:20.475 --> 00:42:22.240 is preserved, of course. 00:42:22.240 --> 00:42:28.800 But this entry over here is simply addition. 00:42:28.800 --> 00:42:31.860 So this entry here is just addition. 00:42:31.860 --> 00:42:34.140 This entry over here is also addition. 00:42:37.040 --> 00:42:40.870 And the top right entry is a bit more complicated. 00:42:40.870 --> 00:42:45.100 It's some addition, but there's an additional twist. 00:42:52.230 --> 00:42:53.900 So this is how matrix multiplication 00:42:53.900 --> 00:42:55.642 works in this group. 00:42:55.642 --> 00:42:57.350 I mean, this is how matrix multiplication 00:42:57.350 --> 00:43:00.530 works, but in terms of elements of this group, that's 00:43:00.530 --> 00:43:02.400 what happens. 00:43:02.400 --> 00:43:05.300 So you see it's kind of like an abelian group, 00:43:05.300 --> 00:43:09.720 but there's an extra twist, so it's almost abelian, so 00:43:09.720 --> 00:43:13.990 the first step you can take away from abelian. 00:43:13.990 --> 00:43:16.220 And there's a way to quantify this notion. 00:43:16.220 --> 00:43:17.470 It's called "nilpotency." 00:43:17.470 --> 00:43:20.040 And we'll get to that in a second. 00:43:20.040 --> 00:43:25.070 But in particular, if you set S to be the following 00:43:25.070 --> 00:43:25.850 generators-- 00:43:32.490 --> 00:43:35.290 so if you take S to be these four elements, 00:43:35.290 --> 00:43:40.720 and you ask what does the r-th power of S look like, 00:43:40.720 --> 00:43:42.250 so I look at all the elements which 00:43:42.250 --> 00:43:48.010 can be written by r or at most r elements from S, 00:43:48.010 --> 00:43:49.960 what do these elements look like? 00:43:54.410 --> 00:43:55.350 What do you think? 00:43:55.350 --> 00:44:01.420 So if you look at elements in here, 00:44:01.420 --> 00:44:06.232 how large can this entry, the 1, comma, 2 entry be? 00:44:06.232 --> 00:44:07.080 r. 00:44:07.080 --> 00:44:10.140 So each time you do addition, so it's at most r. 00:44:10.140 --> 00:44:13.090 So let me be a bit rough here, and say it's big O of r. 00:44:13.090 --> 00:44:16.630 And likewise, the 2, 1, 2, 3, entry is also big O of r. 00:44:16.630 --> 00:44:19.660 What about the top right entry over here? 00:44:23.240 --> 00:44:25.380 So it grows like r squared, because there is 00:44:25.380 --> 00:44:27.170 an extra multiplication term. 00:44:29.790 --> 00:44:33.060 So you can be much more precise about the growth rate 00:44:33.060 --> 00:44:35.320 of these individual entries. 00:44:35.320 --> 00:44:39.150 But very roughly, it looks like this ball over here. 00:44:39.150 --> 00:44:47.230 So the size of S, the r-th ball of S, 00:44:47.230 --> 00:44:52.760 is roughly, it's on the order of 4th power of r. 00:44:55.650 --> 00:45:02.348 So in particular, the doubling constant, 00:45:02.348 --> 00:45:05.960 if r is reasonably large, is what? 00:45:10.440 --> 00:45:12.765 What happens when we go from r to 2r? 00:45:12.765 --> 00:45:16.185 The size increases by a factor of around 16. 00:45:21.610 --> 00:45:26.340 So that's an example of a set in a non-abelian group 00:45:26.340 --> 00:45:28.900 with bounded doubling, which is genuinely 00:45:28.900 --> 00:45:31.410 different from the examples we have seen so far. 00:45:31.410 --> 00:45:32.974 So that's non-abelian. 00:45:32.974 --> 00:45:33.474 Yeah. 00:45:33.474 --> 00:45:37.370 AUDIENCE: [INAUDIBLE] 00:45:37.370 --> 00:45:41.220 YUFEI ZHAO: The question is, is the size-- 00:45:41.220 --> 00:45:44.740 we've shown the size is-- 00:45:44.740 --> 00:45:47.280 I'm not being very precise here, but you can 00:45:47.280 --> 00:45:48.790 do upper bound and lower bound. 00:45:48.790 --> 00:45:51.810 So size turns out to be the order of r to the 4. 00:45:55.538 --> 00:45:57.330 So you want to show that there are actually 00:45:57.330 --> 00:45:59.290 enough elements over here that you can fill in, 00:45:59.290 --> 00:46:00.770 but I'll leave that to you. 00:46:08.218 --> 00:46:10.010 Can you build other examples like this one? 00:46:13.370 --> 00:46:14.810 Yeah. 00:46:14.810 --> 00:46:16.240 AUDIENCE: How do we know that this 00:46:16.240 --> 00:46:20.793 isn't similar to a co-set, the direct sum [INAUDIBLE]?? 00:46:20.793 --> 00:46:22.210 YUFEI ZHAO: Question is, how do we 00:46:22.210 --> 00:46:26.050 know this isn't like a co-set sum or a co-set progression? 00:46:26.050 --> 00:46:29.830 For one thing, this is not abelian. 00:46:29.830 --> 00:46:33.970 S, if you multiply entries of S in different orders, 00:46:33.970 --> 00:46:36.830 you get different elements. 00:46:36.830 --> 00:46:40.030 So already in that way, it's different from the examples 00:46:40.030 --> 00:46:41.135 that we have seen before. 00:46:41.135 --> 00:46:42.010 But no, you're right. 00:46:42.010 --> 00:46:44.740 So maybe we can write this a semi-direct product 00:46:44.740 --> 00:46:46.630 in terms of things we have seen before. 00:46:46.630 --> 00:46:49.600 And it is, in some sense, a semi-direct product, 00:46:49.600 --> 00:46:51.970 but it's a very special kind of semi-direct product. 00:46:58.202 --> 00:47:00.660 From that example, you can build bigger examples, of course 00:47:00.660 --> 00:47:03.060 with more entries in the matrix. 00:47:03.060 --> 00:47:06.480 But more generally, these things are 00:47:06.480 --> 00:47:09.270 what are known as "nilpotent groups." 00:47:09.270 --> 00:47:17.970 So that's an example of a nilpotent group. 00:47:17.970 --> 00:47:20.430 And to remind you, the definition of a nilpotent group 00:47:20.430 --> 00:47:23.700 is a group where the lower central series eventually 00:47:23.700 --> 00:47:24.360 terminates. 00:47:32.790 --> 00:47:36.500 In particular, inside that if you look at-- 00:47:36.500 --> 00:47:41.100 so this is the commutator of G, so look at all the elements 00:47:41.100 --> 00:47:44.208 that we recognize x, y, x inverse, 00:47:44.208 --> 00:47:46.750 y inverse-- the set of elements that can be written this way. 00:47:46.750 --> 00:47:48.240 So that's a subgroup. 00:47:48.240 --> 00:47:57.820 And if I repeat this operation enough times, 00:47:57.820 --> 00:48:02.340 I eventually would get just the identity. 00:48:02.340 --> 00:48:03.940 And you could trade on that group. 00:48:03.940 --> 00:48:09.150 If you do the commutator, so essentially you get rid 00:48:09.150 --> 00:48:15.630 of abelian-ness and you move up the whole diagonal, 00:48:15.630 --> 00:48:19.630 you create a commutator, you'd get rid of these-- 00:48:19.630 --> 00:48:20.650 all these two entries. 00:48:20.650 --> 00:48:23.790 So you get z alone. 00:48:23.790 --> 00:48:26.095 If you do it one more time, you zero out that entry. 00:48:33.710 --> 00:48:39.930 And so more generally, all of these nilpotent groups 00:48:39.930 --> 00:48:49.680 have this phenomenon, have the polynomial growth phenomenon. 00:48:55.180 --> 00:49:00.220 So if you take a set of generators and look at a ball, 00:49:00.220 --> 00:49:01.680 and look at the volume of the ball, 00:49:01.680 --> 00:49:04.210 how does the volume of the ball grow with the radius? 00:49:04.210 --> 00:49:07.160 It grows like a polynomial. 00:49:07.160 --> 00:49:09.250 And so let me define that. 00:49:09.250 --> 00:49:21.330 So given G, a finitely generated group, so generated by set S, 00:49:21.330 --> 00:49:36.560 we say that G has polynomial growth if the size S to the r 00:49:36.560 --> 00:49:40.350 grows like at most a polynomial in r. 00:49:48.470 --> 00:49:50.990 It's worth noting that this definition is really 00:49:50.990 --> 00:49:54.740 a definition about G. It does not depend 00:49:54.740 --> 00:49:56.487 on the choice of generators. 00:50:02.340 --> 00:50:04.790 You can have different choices, generators for the group. 00:50:04.790 --> 00:50:07.640 But if it has polynomial growth with respect 00:50:07.640 --> 00:50:10.210 to one set of generators, then it's the same. 00:50:10.210 --> 00:50:12.260 It also has polynomial growth with regards 00:50:12.260 --> 00:50:13.820 to every other set. 00:50:13.820 --> 00:50:18.970 So we've seen an example of groups with polynomial growth. 00:50:18.970 --> 00:50:21.118 Abelian groups have polynomial growth. 00:50:21.118 --> 00:50:22.660 So if you think of polynomial growth, 00:50:22.660 --> 00:50:25.690 think lattice or z to the m. 00:50:25.690 --> 00:50:27.930 So if you take a ball growing, so it 00:50:27.930 --> 00:50:34.480 has size growing like r to the dimension. 00:50:34.480 --> 00:50:36.600 But nilpotent groups is another example 00:50:36.600 --> 00:50:39.360 of groups with polynomial growth. 00:50:39.360 --> 00:50:42.600 And these are, intuitively at least for now, related 00:50:42.600 --> 00:50:44.707 to bounded doubling. 00:50:44.707 --> 00:50:47.040 If it's polynomial growth, then it has bounded doubling. 00:50:49.580 --> 00:50:52.840 So is there a classification of groups with bounded-- 00:50:52.840 --> 00:50:54.740 with polynomial growth? 00:50:54.740 --> 00:50:57.400 So if I tell you a group-- so an infinite group always, 00:50:57.400 --> 00:51:00.400 because otherwise if finite, then it maxes out already 00:51:00.400 --> 00:51:01.120 at some point. 00:51:01.120 --> 00:51:03.530 So I give you an infinite group. 00:51:03.530 --> 00:51:05.890 I tell you it has polynomial growth. 00:51:05.890 --> 00:51:07.648 What can you tell me about this group? 00:51:07.648 --> 00:51:09.190 Is there some characterization that's 00:51:09.190 --> 00:51:12.133 an inverse of what we've seen so far? 00:51:12.133 --> 00:51:13.050 And the answer is yes. 00:51:13.050 --> 00:51:16.060 And this is a famous and deep result of Gromov. 00:51:19.290 --> 00:51:25.180 So Gromov's theorem on groups of polynomial growth 00:51:25.180 --> 00:51:26.660 from the '80s. 00:51:26.660 --> 00:51:33.910 Gromov showed that a finitely generated 00:51:33.910 --> 00:51:42.400 group has polynomial growth if and only 00:51:42.400 --> 00:51:55.230 if it's virtually nilpotent, where "virtually" 00:51:55.230 --> 00:52:01.200 is an adverb in group theory where 00:52:01.200 --> 00:52:04.950 you have some property like "abelian," or "solvable," 00:52:04.950 --> 00:52:06.030 or whatever. 00:52:06.030 --> 00:52:16.490 So virtually P means that there exists a finite index subgroup 00:52:16.490 --> 00:52:24.950 with property P. So "virtually nilpotent" 00:52:24.950 --> 00:52:29.100 means there is a finite index subgroup that is nilpotent. 00:52:29.100 --> 00:52:34.140 So it completely characterizes groups of polynomial growth. 00:52:34.140 --> 00:52:37.080 So basically, all the examples we've seen so far 00:52:37.080 --> 00:52:43.680 are representative, so up to changing by a finite index 00:52:43.680 --> 00:52:45.780 subgroup, which as you would expect, 00:52:45.780 --> 00:52:50.520 shouldn't change the growth nature by so much. 00:52:50.520 --> 00:52:53.730 There are some analogies to be made here with, 00:52:53.730 --> 00:52:59.130 for example in geometry, you ask in Euclidean space, 00:52:59.130 --> 00:53:03.560 how fast is the ball of radius r growing? 00:53:03.560 --> 00:53:07.580 In dimension d, it grows like r to the d. 00:53:07.580 --> 00:53:10.920 What about in the hyperbolic space? 00:53:10.920 --> 00:53:13.530 Does anyone know how fast, in a hyperbolic space, 00:53:13.530 --> 00:53:15.570 a ball of radius r grows? 00:53:19.130 --> 00:53:23.100 It's exponential in the radius. 00:53:23.100 --> 00:53:27.390 So for non-negatively curved spaces, 00:53:27.390 --> 00:53:30.240 the balls grow polynomially. 00:53:30.240 --> 00:53:33.720 But for something that's negatively curvatured, 00:53:33.720 --> 00:53:35.970 in particular the hyperbolic space, 00:53:35.970 --> 00:53:39.940 the ball growth might be exponential. 00:53:39.940 --> 00:53:43.190 You have a similar phenomenon happening here. 00:53:43.190 --> 00:53:44.870 The opposite of polynomial growth 00:53:44.870 --> 00:53:46.400 is, well, super polynomial growth, 00:53:46.400 --> 00:53:53.050 but one specific example is that of a free group, where 00:53:53.050 --> 00:53:56.730 there are no relations between the generators. 00:53:56.730 --> 00:54:03.930 In that case, the balls, they grow like exponentially. 00:54:03.930 --> 00:54:08.550 So the balls grow exponentially in the radius. 00:54:08.550 --> 00:54:11.470 Gromov's theorem is a deep theorem. 00:54:11.470 --> 00:54:16.150 And its original proof used some very hard tools 00:54:16.150 --> 00:54:17.830 coming from geometry. 00:54:17.830 --> 00:54:23.230 And Gromov developed a notion of convergence of metric spaces, 00:54:23.230 --> 00:54:26.740 somewhat akin to our discussion of graph limits. 00:54:26.740 --> 00:54:28.960 So starting with discrete objects, 00:54:28.960 --> 00:54:32.890 he looked at some convergence to some continuous objects, 00:54:32.890 --> 00:54:37.420 and then used some very deep results 00:54:37.420 --> 00:54:41.740 from the classification of locally compact groups 00:54:41.740 --> 00:54:44.410 to derive this result over here. 00:54:48.060 --> 00:54:50.840 So this proof has been quite influential, 00:54:50.840 --> 00:54:53.930 and is related to something called 00:54:53.930 --> 00:55:05.640 "Hilbert's fifth problem, which concerns characterizations 00:55:05.640 --> 00:55:07.336 of Lie groups. 00:55:07.336 --> 00:55:09.620 So all of these are inverse-type problems. 00:55:09.620 --> 00:55:11.690 I tell you some structure has some property. 00:55:11.690 --> 00:55:15.770 Describe that structure. 00:55:15.770 --> 00:55:20.030 What does this all have to do with Freiman's theorem? 00:55:20.030 --> 00:55:21.310 Already you see some relation. 00:55:21.310 --> 00:55:23.602 So there seems, at least intuitively, some relationship 00:55:23.602 --> 00:55:26.270 between groups of polynomial growth versus subsets 00:55:26.270 --> 00:55:27.650 of bounded doubling. 00:55:27.650 --> 00:55:31.320 One implies the other, although not in the converse. 00:55:31.320 --> 00:55:32.570 And they are indeed related. 00:55:32.570 --> 00:55:35.720 And this comes out of some very recent work. 00:55:35.720 --> 00:55:37.790 I should also mention that Gromov's theorem has 00:55:37.790 --> 00:55:40.730 been made simplified by Kleiner, who 00:55:40.730 --> 00:55:44.930 gave an important simplification, a more 00:55:44.930 --> 00:55:47.080 elementary proof of Gromov's theorem. 00:55:49.790 --> 00:55:52.160 So let's talk about the non-abelian version 00:55:52.160 --> 00:55:53.330 of Freiman's theorem. 00:55:59.630 --> 00:56:02.270 We would like some result that says 00:56:02.270 --> 00:56:10.220 that is it true that every set, most every set of-- 00:56:10.220 --> 00:56:12.705 so previously, we had small doubling. 00:56:12.705 --> 00:56:15.080 You want to have some similar notion, although it may not 00:56:15.080 --> 00:56:19.330 be exactly small doubling, but let me not be very precise 00:56:19.330 --> 00:56:24.043 and to say, "small doubling." 00:56:24.043 --> 00:56:26.210 In literature, these things are sometimes also known 00:56:26.210 --> 00:56:27.770 as "approximate groups." 00:56:32.027 --> 00:56:34.930 So if you look this up, you will get to the relevant literature 00:56:34.930 --> 00:56:36.250 on the subject. 00:56:36.250 --> 00:56:39.160 Most every set of small doubling in some non-abelian group 00:56:39.160 --> 00:56:45.670 behaves like one of these known examples, something 00:56:45.670 --> 00:56:58.160 which is some combination of subgroups and nilpotent balls. 00:57:03.300 --> 00:57:06.240 So these combinations are sometimes known as "co-set 00:57:06.240 --> 00:57:07.729 nilprogressions." 00:57:15.560 --> 00:57:18.910 So this was something that was only 00:57:18.910 --> 00:57:21.190 explored in the past 10 years or so 00:57:21.190 --> 00:57:25.540 in a series of very difficult works. 00:57:25.540 --> 00:57:27.940 Previously, it had been known, and still 00:57:27.940 --> 00:57:30.190 was being investigated for various special classes 00:57:30.190 --> 00:57:32.800 of matrix groups or special classes of groups 00:57:32.800 --> 00:57:35.850 like solvable groups and whatnot, 00:57:35.850 --> 00:57:38.900 that are more explicit or easier to handle 00:57:38.900 --> 00:57:40.730 or closer to the abelian analog. 00:57:43.580 --> 00:57:50.370 There was important work of Hrushovski, 00:57:50.370 --> 00:57:55.090 which was published about 10 years ago, 00:57:55.090 --> 00:57:58.320 who showed using model theory techniques, 00:57:58.320 --> 00:58:05.700 so using methods from logic, that a weak version 00:58:05.700 --> 00:58:12.970 of Freiman's theorem is true for non-abelian groups. 00:58:12.970 --> 00:58:22.842 And later on, Breuillard, Green, and Tao building on Hrushovskis 00:58:22.842 --> 00:58:24.800 work-- so this actually came quite a bit later, 00:58:24.800 --> 00:58:26.730 even though the journal publication dates are the same 00:58:26.730 --> 00:58:27.380 year-- 00:58:27.380 --> 00:58:31.430 so they were able to build on Hrushovski's work, 00:58:31.430 --> 00:58:35.480 and greatly expanding on it, and going back to some of the older 00:58:35.480 --> 00:58:40.070 techniques coming from Hilbert's fifth problem, and as a result, 00:58:40.070 --> 00:58:43.580 proved an inverse structure theorem 00:58:43.580 --> 00:58:46.760 that gave some kind of answer to this question 00:58:46.760 --> 00:58:48.830 of non-abelian Freiman. 00:58:48.830 --> 00:58:51.380 So we now do have some theorem which 00:58:51.380 --> 00:58:54.320 is like Freiman's theorem for abelian groups that 00:58:54.320 --> 00:58:57.560 says in a non-abelian group, if you have something that 00:58:57.560 --> 00:59:02.060 resembles small doubling, then the set must, in some sense, 00:59:02.060 --> 00:59:06.680 look like a combination of subgroups and nilpotent balls. 00:59:06.680 --> 00:59:09.350 But let me not be precise at all. 00:59:09.350 --> 00:59:10.940 The methods here build on Hrushovski. 00:59:10.940 --> 00:59:14.930 And Hrushovski used model theory, which is kind of-- 00:59:14.930 --> 00:59:19.640 it's something where-- in particular, 00:59:19.640 --> 00:59:21.560 one feature of all of these proofs 00:59:21.560 --> 00:59:22.910 is that they give no bounds. 00:59:26.950 --> 00:59:29.650 Similar to what we've seen earlier in the course, 00:59:29.650 --> 00:59:33.790 in proofs that involved compactness, what happens 00:59:33.790 --> 00:59:36.055 here is that the arguments use ultra filters. 00:59:39.420 --> 00:59:43.190 So there are these constructions from mathematical logic. 00:59:43.190 --> 00:59:46.340 And like compactness, they give no bounds. 00:59:46.340 --> 00:59:49.720 So it remains an open problem to prove Freiman's theorem 00:59:49.720 --> 00:59:52.420 for non-abelian groups with some concrete bounds. 00:59:52.420 --> 00:59:53.197 Question. 00:59:53.197 --> 00:59:57.230 AUDIENCE: [INAUDIBLE] nilpotent ball? 00:59:57.230 --> 00:59:59.480 YUFEI ZHAO: What is nilpotent ball? 00:59:59.480 --> 01:00:01.610 I don't want to give a precise definition, 01:00:01.610 --> 01:00:04.610 but roughly speaking, it's balls that 01:00:04.610 --> 01:00:07.790 come out of those types of constructions. 01:00:07.790 --> 01:00:09.610 So you take a nilpotent subgroup. 01:00:09.610 --> 01:00:11.030 You take a nilpotent group. 01:00:11.030 --> 01:00:13.400 You look at an image of a nilpotent group 01:00:13.400 --> 01:00:23.640 into your group, and then look at the image of that ball, so 01:00:23.640 --> 01:00:29.130 something that looks like one of the previous constructions. 01:00:29.130 --> 01:00:31.495 So that's all I want to say about non-abelian extensions 01:00:31.495 --> 01:00:32.370 of Freiman's theorem. 01:00:32.370 --> 01:00:33.598 Any questions? 01:00:33.598 --> 01:00:35.140 AUDIENCE: Would you say one more time 01:00:35.140 --> 01:00:37.408 what you mean by "approximate group?" 01:00:37.408 --> 01:00:38.700 YUFEI ZHAO: So what I mean by-- 01:00:38.700 --> 01:00:41.670 you can look in the papers and see the precise definitions, 01:00:41.670 --> 01:00:47.700 but roughly speaking, it's that if you have-- 01:00:47.700 --> 01:00:51.090 there are different kinds of definitions and most of them 01:00:51.090 --> 01:00:51.930 are equivalent. 01:00:51.930 --> 01:00:54.120 But one version is that you have a set A 01:00:54.120 --> 01:01:05.940 such that A is coverable by K translates of A, 01:01:05.940 --> 01:01:08.760 so it's a bit more than just the size information, 01:01:08.760 --> 01:01:11.770 but it's actually related to size information. 01:01:11.770 --> 01:01:16.060 So we've already seen in this course how many of these 01:01:16.060 --> 01:01:18.060 different notions can go back and forth from one 01:01:18.060 --> 01:01:21.040 to the other, covering to size, and whatnot. 01:01:27.110 --> 01:01:28.880 The final thing I want to discuss today 01:01:28.880 --> 01:01:31.780 is one of the most central open problems 01:01:31.780 --> 01:01:35.230 in additive combinatorics going back to the abelian version. 01:01:35.230 --> 01:01:37.930 So this is known as the "polynomial Freiman-Ruzsa 01:01:37.930 --> 01:01:38.933 conjecture." 01:01:48.600 --> 01:01:52.050 So we would like some kind of a Freiman theorem 01:01:52.050 --> 01:01:56.370 that preserves the constants up to polynomial changes 01:01:56.370 --> 01:01:59.788 without losing an exponent. 01:01:59.788 --> 01:02:02.980 Now, from earlier discussions, I showed you 01:02:02.980 --> 01:02:09.950 that the bounds that we almost proved is close to the truth. 01:02:09.950 --> 01:02:11.710 You do need some kind of exponential loss 01:02:11.710 --> 01:02:14.975 in the blowup size of the GAP. 01:02:14.975 --> 01:02:16.600 But it turns out those kind of examples 01:02:16.600 --> 01:02:18.100 are slightly misleading. 01:02:18.100 --> 01:02:20.800 So let's look at the examples of the constructions again. 01:02:20.800 --> 01:02:26.500 So if A-- so just for simplicity in exposition, 01:02:26.500 --> 01:02:31.550 I'm going to stick with F2 to the n, at least initially. 01:02:31.550 --> 01:02:41.090 So if A is an independent set of size n, 01:02:41.090 --> 01:02:45.660 then K, being the doubling constant of A, 01:02:45.660 --> 01:02:48.650 is roughly like n over 2. 01:02:48.650 --> 01:02:52.145 And yet the subgroup that contains A 01:02:52.145 --> 01:03:00.110 has size 2 to the something on the order of K times A. 01:03:00.110 --> 01:03:04.830 So you necessarily incur an exponential loss over here. 01:03:04.830 --> 01:03:08.740 Now, you might complain that the size of A here is basically K. 01:03:08.740 --> 01:03:11.570 But of course, I can blow up this example 01:03:11.570 --> 01:03:17.600 by considering what happens if you take each element here, 01:03:17.600 --> 01:03:19.610 and blow it up into an entire subspace. 01:03:25.110 --> 01:03:28.150 So the e's are the coordinate vectors. 01:03:28.150 --> 01:03:32.340 So now I'm sitting inside F2 to the m plus n. 01:03:32.340 --> 01:03:33.690 And that gives me this set. 01:03:36.800 --> 01:03:39.220 The doubling constant is still the same as before. 01:03:44.900 --> 01:03:54.270 And yet, we see that the subgroup generated by A 01:03:54.270 --> 01:03:56.310 still has this exponential blowup 01:03:56.310 --> 01:04:00.630 in this constant, exponential in the doubling constant. 01:04:00.630 --> 01:04:04.050 But now you see in this example here, 01:04:04.050 --> 01:04:07.320 even though the subgroup generated by A 01:04:07.320 --> 01:04:11.190 can be much larger than A, so everything's still constant, so 01:04:11.190 --> 01:04:14.100 much larger in terms of as a function of the doubling 01:04:14.100 --> 01:04:18.740 constant, A has a very large structure. 01:04:18.740 --> 01:04:25.070 So A contains a very large subspace. 01:04:30.625 --> 01:04:32.685 By "subspace," I mean affine subspace. 01:04:36.300 --> 01:04:40.920 And the subspace here is comparable to the size 01:04:40.920 --> 01:04:41.850 of A itself. 01:04:44.900 --> 01:04:47.980 So you might wonder, if you don't 01:04:47.980 --> 01:04:52.540 care about containing A inside a single subspace, 01:04:52.540 --> 01:04:56.920 can you do much better in terms of bounds? 01:04:56.920 --> 01:04:59.170 And that's the content of the polynomial Freiman-Ruzsa 01:04:59.170 --> 01:05:01.030 conjecture. 01:05:01.030 --> 01:05:07.030 The PFR conjecture for F2 to the m 01:05:07.030 --> 01:05:13.609 says that if you have a subset of F2 to the m 01:05:13.609 --> 01:05:21.010 and A plus A is size at most K times the size of A, 01:05:21.010 --> 01:05:28.840 then there exists a subspace V of size 01:05:28.840 --> 01:05:44.880 at most A such that V contains a large proportion of A. 01:05:44.880 --> 01:05:46.350 And the large here-- 01:05:46.350 --> 01:05:50.220 we only lose something that is polynomial in these doubling 01:05:50.220 --> 01:05:51.162 constants. 01:05:54.120 --> 01:05:55.540 So that's the case. 01:05:55.540 --> 01:05:57.300 It's over here. 01:05:57.300 --> 01:06:04.160 So instead of containing A inside an entire subspace, 01:06:04.160 --> 01:06:07.850 I just want to contain a large fraction of A in a subspace. 01:06:07.850 --> 01:06:09.740 And the conjecture is that I do not 01:06:09.740 --> 01:06:12.725 need to incur exponential losses in the constants. 01:06:12.725 --> 01:06:15.520 AUDIENCE: So V is an affine subspace? 01:06:15.520 --> 01:06:18.390 YUFEI ZHAO: V is-- 01:06:18.390 --> 01:06:20.552 question is, V is an affine subspace. 01:06:20.552 --> 01:06:22.260 You can think of V as an affine subspace. 01:06:22.260 --> 01:06:23.910 You can think of V as a subspace. 01:06:23.910 --> 01:06:26.179 It doesn't actually matter in this formulation. 01:06:35.170 --> 01:06:36.860 There's an equivalent formulation 01:06:36.860 --> 01:06:41.660 which you might like better, where you might complain, 01:06:41.660 --> 01:06:43.382 initially, PFR is initially-- 01:06:43.382 --> 01:06:44.840 Freiman's theorem is about covering 01:06:44.840 --> 01:06:49.005 A. And now we've only covered a part of A. 01:06:49.005 --> 01:06:50.880 But of course, we saw from earlier arguments, 01:06:50.880 --> 01:06:53.160 you can use Ruzsa's covering lemma to go 01:06:53.160 --> 01:06:56.890 from covering a part of A to covering all of A. 01:06:56.890 --> 01:07:00.370 Indeed, it this the case that this formulation 01:07:00.370 --> 01:07:04.240 is equivalent to the formulation that if A 01:07:04.240 --> 01:07:14.300 is in F to the n and A plus A size at most K times A, 01:07:14.300 --> 01:07:23.150 then there exists some subspace V with the size of V no larger 01:07:23.150 --> 01:07:28.010 than the size of A, such that A can 01:07:28.010 --> 01:07:43.100 be covered by polynomial in K many co-sets of V. 01:07:43.100 --> 01:07:46.130 We see that here. 01:07:46.130 --> 01:07:52.790 Here A has doubling constant K, which is around the same as n. 01:07:52.790 --> 01:07:58.760 And even though I cannot contain A by a single subspace 01:07:58.760 --> 01:08:04.710 of roughly the same size, I can use K different translates 01:08:04.710 --> 01:08:14.440 to cover A. Any questions? 01:08:19.340 --> 01:08:22.460 So I want to leave it to you as an exercise 01:08:22.460 --> 01:08:27.790 to prove that these two versions are equivalent to each other. 01:08:27.790 --> 01:08:29.154 It's not too hard. 01:08:29.154 --> 01:08:31.510 It's something if I had more time, I would show you. 01:08:31.510 --> 01:08:37.279 It uses Ruzsa covering lemma to prove this equivalence. 01:08:39.859 --> 01:08:41.220 The nice thing about the-- 01:08:41.220 --> 01:08:44.180 so the polynomial Freiman-Ruzsa conjecture, PFR conjecture, 01:08:44.180 --> 01:08:46.609 is considered a central conjecture 01:08:46.609 --> 01:08:51.140 in additive combinatorics, because it has many equivalent 01:08:51.140 --> 01:08:54.590 formulations and relates to many problems that 01:08:54.590 --> 01:08:56.990 are central to the subject. 01:08:56.990 --> 01:08:59.720 So we would like some kind of an inverse theorem that gives you 01:08:59.720 --> 01:09:01.220 these polynomial bounds. 01:09:01.220 --> 01:09:06.640 And I'll mention a couple of these equivalent formulations. 01:09:06.640 --> 01:09:10.200 Here is an equivalent formulation 01:09:10.200 --> 01:09:16.700 which is rather attractive, where instead of considering 01:09:16.700 --> 01:09:18.859 subsets, we're going to formulate 01:09:18.859 --> 01:09:22.354 something that has to do with approximate homomorphisms. 01:09:27.310 --> 01:09:32.010 So the statement still conjecture 01:09:32.010 --> 01:09:37.950 is that if F is a function from a Boolean space 01:09:37.950 --> 01:09:45.380 to another Boolean space is such that F is approximately 01:09:45.380 --> 01:09:52.727 a homomorphism in the sense that the set of possible errors-- 01:09:52.727 --> 01:09:54.480 so if it's actually a homomorphism, 01:09:54.480 --> 01:09:56.460 then this quantity is always equal to 0-- 01:09:56.460 --> 01:09:59.760 but it's approximately a homomorphism in the sense 01:09:59.760 --> 01:10:11.140 that the set of such errors is bounded by K in size, 01:10:11.140 --> 01:10:14.730 the conclusion, the conjecture claims that then there 01:10:14.730 --> 01:10:24.650 exists an actual homomorphism, an actual linear map G, 01:10:24.650 --> 01:10:32.810 such that F is very close to G, as 01:10:32.810 --> 01:10:40.450 in that the set of possible discrepancies between F and G 01:10:40.450 --> 01:10:52.890 is bounded, where you only lose at most a polynomial in K. 01:10:52.890 --> 01:10:55.600 So if you are an approximate homomorphism in this sense, 01:10:55.600 --> 01:11:00.500 then you are actually very close to an actual linear map. 01:11:00.500 --> 01:11:06.050 Now, it is not too hard to prove a much quantitatively weaker 01:11:06.050 --> 01:11:07.580 version of this statement. 01:11:07.580 --> 01:11:15.985 So I claim that it is trivial to show upper bound of at most 2 01:11:15.985 --> 01:11:20.860 to the K over here. 01:11:20.860 --> 01:11:22.300 So think about that. 01:11:22.300 --> 01:11:25.390 So if I give you an F, I can just 01:11:25.390 --> 01:11:30.050 think about what the values of F are on the basis, 01:11:30.050 --> 01:11:32.660 and extend it to a linear map. 01:11:35.570 --> 01:11:44.160 Then this set is necessarily a span of that set, 01:11:44.160 --> 01:11:51.150 so has size at most 2 to the K. But it's open to show you only 01:11:51.150 --> 01:12:01.870 have to lose a polynomial in K. 01:12:01.870 --> 01:12:06.890 There is also a version of the polynomial Freiman-Ruzsa 01:12:06.890 --> 01:12:12.500 conjecture which is related to things we've discussed earlier 01:12:12.500 --> 01:12:15.970 regarding Szemeredi's theorem. 01:12:15.970 --> 01:12:20.680 And in fact, the polynomial Freiman-Ruzsa conjecture kind 01:12:20.680 --> 01:12:24.820 of came back into popularity partly because 01:12:24.820 --> 01:12:28.400 of Gowers' proof of Szemeredi's theorem 01:12:28.400 --> 01:12:31.260 that used many of these tools. 01:12:31.260 --> 01:12:34.250 So let me state it here. 01:12:34.250 --> 01:12:41.320 So we've seen some statement like this in an earlier 01:12:41.320 --> 01:12:48.670 lecture, but not very precisely or not precisely in this form. 01:12:48.670 --> 01:12:53.600 And I won't define for you all the notation here, 01:12:53.600 --> 01:12:57.050 but hopefully, you get a rough sense of what it's about. 01:12:57.050 --> 01:12:59.200 So we want some kind of an inverse statement 01:12:59.200 --> 01:13:03.670 for what's known as a "quadratic uniformity norm," 01:13:03.670 --> 01:13:05.380 "quadratic Gowers' uniformity norm." 01:13:12.700 --> 01:13:17.020 So recall back to our discussion of the proof of Roth's theorem, 01:13:17.020 --> 01:13:20.050 the Fourier analytic proof of Roth's theorem. 01:13:20.050 --> 01:13:21.820 We want to say that-- 01:13:21.820 --> 01:13:26.590 but now think about not three APs, but four APs. 01:13:26.590 --> 01:13:32.910 So we want to know if you have a function F on the Boolean cube, 01:13:32.910 --> 01:13:43.667 and this function is 1 bounded, and-- 01:13:43.667 --> 01:13:45.500 I'm going to write down some notation, which 01:13:45.500 --> 01:13:47.190 we are not going to define-- 01:13:47.190 --> 01:13:54.880 but the Gowers' u3 norm is at least some delta. 01:13:54.880 --> 01:13:58.570 So this is something which is related to 4 AP counts. 01:13:58.570 --> 01:14:00.868 So in particular, if this number is small, 01:14:00.868 --> 01:14:03.160 then you have a counting lemma for four-term arithmetic 01:14:03.160 --> 01:14:06.510 progressions. 01:14:06.510 --> 01:14:17.330 If this is true, then there exists a quadratic polynomial q 01:14:17.330 --> 01:14:27.520 in n variables over F2 such that your function 01:14:27.520 --> 01:14:34.660 F correlates with this quadratic exponential in q. 01:14:34.660 --> 01:14:44.750 And the correlation here is something where you only lose 01:14:44.750 --> 01:14:48.320 a polynomial in the parameters. 01:14:48.320 --> 01:14:49.830 So previously, I quoted something 01:14:49.830 --> 01:14:55.410 where you lose something that's only a constant in delta, 01:14:55.410 --> 01:14:56.400 and that is true. 01:14:56.400 --> 01:14:57.410 That is known. 01:14:57.410 --> 01:15:00.940 But we believe, so it's conjecture, 01:15:00.940 --> 01:15:03.650 that you only lose a polynomial in these parameters. 01:15:03.650 --> 01:15:05.300 So this type of statement-- remember, 01:15:05.300 --> 01:15:07.800 in our proof of Roth's theorem, something like this came up. 01:15:07.800 --> 01:15:10.120 So something like this came up as a crucial step 01:15:10.120 --> 01:15:11.440 in the proof of Roth's theorem. 01:15:11.440 --> 01:15:17.680 If you have something where you look at counting lemma, 01:15:17.680 --> 01:15:19.180 and you exhibit something like this, 01:15:19.180 --> 01:15:22.110 then you can exhibit a large Fourier character. 01:15:22.110 --> 01:15:24.130 And in higher order Fourier analysis, 01:15:24.130 --> 01:15:27.850 something like this corresponds to having 01:15:27.850 --> 01:15:29.541 a large Fourier transform. 01:15:32.310 --> 01:15:34.440 It turns out that all of these formulations 01:15:34.440 --> 01:15:36.540 of polynomial Freiman-Ruzsa conjecture 01:15:36.540 --> 01:15:40.410 are equivalent to each other. 01:15:40.410 --> 01:15:45.420 And they're all equivalent in a very quantitative sense, 01:15:45.420 --> 01:15:53.840 so up to polynomial changes in the bounds. 01:15:57.563 --> 01:15:58.980 So in particular, if you can prove 01:15:58.980 --> 01:16:01.230 some bound for some version, then that automatically 01:16:01.230 --> 01:16:04.320 leads to bounds for the other versions. 01:16:04.320 --> 01:16:06.870 The proof of equivalences is not trivial, 01:16:06.870 --> 01:16:12.210 but it's also not too complicated. 01:16:12.210 --> 01:16:15.630 It takes some work, but it's not too complicated. 01:16:15.630 --> 01:16:19.680 The best bounds for the polynomial Freiman-Ruzsa 01:16:19.680 --> 01:16:22.500 conjecture, and hence for all of these versions, 01:16:22.500 --> 01:16:26.100 is again due to Tom Sanders. 01:16:26.100 --> 01:16:45.360 And he proved a version of PFR with quasi-polynomial bounds, 01:16:45.360 --> 01:16:48.310 where by "quasi-polynomial bounds," I mean, 01:16:48.310 --> 01:16:53.430 for instance over here, instead of K. 01:16:53.430 --> 01:17:02.650 He proved it for something which is like e to the poly log K, 01:17:02.650 --> 01:17:07.980 so like K to the log K, but K to the poly log K. 01:17:07.980 --> 01:17:11.410 So it's almost polynomial, but not quite there. 01:17:14.180 --> 01:17:17.570 And it's considered a central open problem 01:17:17.570 --> 01:17:22.010 to better understand the polynomial Freiman-Ruzsa 01:17:22.010 --> 01:17:24.150 conjecture. 01:17:24.150 --> 01:17:26.010 And we believe that this is something 01:17:26.010 --> 01:17:30.330 that could lead to a lot of important new tools 01:17:30.330 --> 01:17:32.340 and techniques that are relevant to the rest 01:17:32.340 --> 01:17:35.016 of additive combinatorics. 01:17:35.016 --> 01:17:37.330 Yeah. 01:17:37.330 --> 01:17:39.790 AUDIENCE: Using the fact that all of these are equivalent, 01:17:39.790 --> 01:17:43.000 is it possible to get a proof of Freiman's theorem using 01:17:43.000 --> 01:17:46.062 the bound of 2 to the K to be approximate [INAUDIBLE]?? 01:17:46.062 --> 01:17:47.520 YUFEI ZHAO: OK, so the question is, 01:17:47.520 --> 01:17:51.890 we know that that up there has 2 to the K, 01:17:51.890 --> 01:17:56.550 so you're asking can you use this 2 to the K 01:17:56.550 --> 01:18:00.790 to get some bound for polynomial, 01:18:00.790 --> 01:18:03.130 for something like this? 01:18:03.130 --> 01:18:04.380 And the answer is yes. 01:18:04.380 --> 01:18:07.250 So you can use that proof to go through some proofs 01:18:07.250 --> 01:18:09.030 and get here. 01:18:09.030 --> 01:18:12.870 I don't remember how this equivalence goes, but remember 01:18:12.870 --> 01:18:16.920 that the proof of Freiman's theorem for F2 to the n 01:18:16.920 --> 01:18:19.260 wasn't so hard. 01:18:19.260 --> 01:18:22.500 So we didn't use very many tools. 01:18:22.500 --> 01:18:24.870 Unfortunately, I don't have time to tell you 01:18:24.870 --> 01:18:29.070 the formulations of polynomial Freiman-Ruzsa conjecture 01:18:29.070 --> 01:18:33.330 over the integers, and also over arbitrary abelian groups. 01:18:33.330 --> 01:18:36.480 But there are formulations over the integers, 01:18:36.480 --> 01:18:40.140 and that's one that people care just as much about. 01:18:40.140 --> 01:18:42.990 And there are also different equivalent versions, 01:18:42.990 --> 01:18:47.457 but things are a bit nicer in the Boolean case. 01:18:47.457 --> 01:18:48.894 Yeah. 01:18:48.894 --> 01:18:50.331 AUDIENCE: You said [INAUDIBLE]? 01:18:52.496 --> 01:18:54.621 YUFEI ZHAO: I'm sorry, can you repeat the question? 01:18:54.621 --> 01:18:58.389 AUDIENCE: [INAUDIBLE]. 01:18:58.389 --> 01:18:59.472 Yeah, what does that mean? 01:18:59.472 --> 01:19:01.431 YUFEI ZHAO: Are you asking what does this mean? 01:19:01.431 --> 01:19:02.163 AUDIENCE: Yeah. 01:19:02.163 --> 01:19:04.580 YUFEI ZHAO: So this is what's called a "Gowers' uniformity 01:19:04.580 --> 01:19:06.090 norm." 01:19:06.090 --> 01:19:13.820 So something I encourage you to look up. 01:19:13.820 --> 01:19:17.790 In fact, there is an unassigned problem in the problem set 01:19:17.790 --> 01:19:21.150 that's related to the Gowers' uniformity norm 01:19:21.150 --> 01:19:24.610 before you U2, which just relates to Fourier analysis. 01:19:24.610 --> 01:19:29.500 But U3 is related to 4 AP's and quadratic Fourier analysis.