1 00:00:19,195 --> 00:00:19,820 YUFEI ZHAO: OK. 2 00:00:19,820 --> 00:00:22,760 I want to begin by giving some comments regarding 3 00:00:22,760 --> 00:00:23,865 the Wikipedia assignment. 4 00:00:23,865 --> 00:00:26,480 So I sent out an email about this last night. 5 00:00:26,480 --> 00:00:29,510 And so first of all, thank you for your contributions 6 00:00:29,510 --> 00:00:32,280 to this assignment, to Wikipedia. 7 00:00:32,280 --> 00:00:36,410 It plays a really important role in educating a wider audience 8 00:00:36,410 --> 00:00:39,260 what this subject is about because as many of you 9 00:00:39,260 --> 00:00:41,555 have experienced, the first time-- 10 00:00:41,555 --> 00:00:43,520 if you heard of some term, you have 11 00:00:43,520 --> 00:00:45,560 no idea what it is, you put it into Google, 12 00:00:45,560 --> 00:00:49,220 and often Wikipedia's entry is one of the top results that 13 00:00:49,220 --> 00:00:50,120 come up. 14 00:00:50,120 --> 00:00:52,700 And what gets written in there actually 15 00:00:52,700 --> 00:00:56,990 plays a fairly influential role in educating a broader audience 16 00:00:56,990 --> 00:00:59,520 about what this topic is about. 17 00:00:59,520 --> 00:01:04,370 And so I want to emphasize that this is not simply 18 00:01:04,370 --> 00:01:05,930 some homework assignment. 19 00:01:05,930 --> 00:01:08,300 It's something that is a real contribution. 20 00:01:08,300 --> 00:01:10,610 And it's something that contributes 21 00:01:10,610 --> 00:01:13,370 to the dissemination of knowledge. 22 00:01:13,370 --> 00:01:15,860 And for that, it is really important to do a good job, 23 00:01:15,860 --> 00:01:21,020 to do it right, to do it well, so that next time someone-- 24 00:01:21,020 --> 00:01:22,962 maybe even yourselves-- maybe you've 25 00:01:22,962 --> 00:01:24,920 forgotten what the subject is about and go back 26 00:01:24,920 --> 00:01:28,750 and you want to look it up again and remind yourself. 27 00:01:28,750 --> 00:01:32,420 You will have a useful resource to look into. 28 00:01:32,420 --> 00:01:35,450 But also let's say someone wants to find out what 29 00:01:35,450 --> 00:01:38,180 is external graph theory about? 30 00:01:38,180 --> 00:01:40,700 What is additive combinatorics about? 31 00:01:40,700 --> 00:01:43,190 You want them to land on the page that points you 32 00:01:43,190 --> 00:01:46,160 to the right type of places, that points you 33 00:01:46,160 --> 00:01:50,870 to useful resources, that opens doors so that you can explore 34 00:01:50,870 --> 00:01:51,680 further. 35 00:01:51,680 --> 00:01:55,480 And some of the contributions, indeed, 36 00:01:55,480 --> 00:01:57,170 serve you well in that purpose. 37 00:01:57,170 --> 00:01:59,600 It opens doors to many things. 38 00:01:59,600 --> 00:02:01,520 And part of the spirit of this assignment 39 00:02:01,520 --> 00:02:04,010 is for you to do your own research, 40 00:02:04,010 --> 00:02:07,940 do your own literature search, to learn more about a subject, 41 00:02:07,940 --> 00:02:11,060 more than what has been taught in these lectures 42 00:02:11,060 --> 00:02:13,770 so that you can write about it on Wikipedia. 43 00:02:13,770 --> 00:02:15,970 You can link to more references, you 44 00:02:15,970 --> 00:02:20,460 know, show the world what the subject is about. 45 00:02:20,460 --> 00:02:25,130 OK, continuing with our program, so we 46 00:02:25,130 --> 00:02:28,160 spent the past few lectures developing tools regarding 47 00:02:28,160 --> 00:02:31,220 the structure of set addition so that we 48 00:02:31,220 --> 00:02:33,050 can prove Freiman's theorem. 49 00:02:33,050 --> 00:02:35,630 So that's been our goal for the past few lectures. 50 00:02:35,630 --> 00:02:38,570 And today we'll finally prove Freiman's theorem. 51 00:02:38,570 --> 00:02:40,690 But let me first remind you the statement-- 52 00:02:40,690 --> 00:02:43,300 so in Freiman's theorem, we would like to show that if you 53 00:02:43,300 --> 00:02:44,580 are in a subset-- 54 00:02:44,580 --> 00:02:47,780 if you're in the integers, you have a set A that has bounded 55 00:02:47,780 --> 00:02:52,130 doubling-- doubling constant, constant k-- 56 00:02:52,130 --> 00:02:55,430 then the said must be contained in a small, generalized 57 00:02:55,430 --> 00:02:59,120 arithmetic progression, down to dimension and size, 58 00:02:59,120 --> 00:03:03,750 only a constant factor larger than a. 59 00:03:03,750 --> 00:03:08,190 We developed various tools the past three lectures building up 60 00:03:08,190 --> 00:03:09,750 to your intermediate results. 61 00:03:09,750 --> 00:03:12,000 But we also collected this very nice set of tools 62 00:03:12,000 --> 00:03:14,580 for proving Freiman's theorem. 63 00:03:14,580 --> 00:03:16,890 So let me review some of them, which 64 00:03:16,890 --> 00:03:19,890 we'll encounter again today. 65 00:03:19,890 --> 00:03:24,510 PlünneckeRuzsa inequality tells you that if you have a set with 66 00:03:24,510 --> 00:03:27,690 small doubling, then the further iterated sums are also 67 00:03:27,690 --> 00:03:28,860 controlled. 68 00:03:28,860 --> 00:03:33,590 So I want you to think of these parameters as k is a constant, 69 00:03:33,590 --> 00:03:36,040 so k to the some power is still a constant, 70 00:03:36,040 --> 00:03:40,740 but also I don't really care about polynomial changes in k. 71 00:03:40,740 --> 00:03:43,750 So I-- you know, we should ignore polynomial changes in k 72 00:03:43,750 --> 00:03:46,900 and view this constant more or less as the original k itself. 73 00:03:46,900 --> 00:03:51,140 So if some is-- the a plus a is around the same size as a, 74 00:03:51,140 --> 00:03:56,470 then further iterations also do not change the sizes very much. 75 00:03:56,470 --> 00:03:58,470 Ruzsa covering lemma: so this was some statement 76 00:03:58,470 --> 00:04:03,550 that if x plus b looks like it could be covered by copies 77 00:04:03,550 --> 00:04:06,580 of b, just in terms of their sizes alone, then in fact, 78 00:04:06,580 --> 00:04:09,760 x could be covered by a small number of translates 79 00:04:09,760 --> 00:04:11,560 of a slightly larger ball. 80 00:04:11,560 --> 00:04:14,110 But here B can be any set. 81 00:04:14,110 --> 00:04:16,870 We had a thing called Ruzsa modeling lemma. 82 00:04:16,870 --> 00:04:18,370 In particular, a consequence of it 83 00:04:18,370 --> 00:04:21,100 is that if a has small doubling, then there 84 00:04:21,100 --> 00:04:23,290 exists the prime n that's not too much 85 00:04:23,290 --> 00:04:26,450 bigger than the size of a, and a very large proportion-- 86 00:04:26,450 --> 00:04:29,440 an eighth of a subset of an eighth of a such 87 00:04:29,440 --> 00:04:32,080 that this subset a prime is prime 88 00:04:32,080 --> 00:04:34,500 and 8 isomorphic to a subset of z mod n. 89 00:04:34,500 --> 00:04:36,250 So even though you start with a set that's 90 00:04:36,250 --> 00:04:38,350 potentially very spread out, provided 91 00:04:38,350 --> 00:04:40,990 they have small doubling, I can pick out 92 00:04:40,990 --> 00:04:43,540 a pretty large piece of it and model it 93 00:04:43,540 --> 00:04:46,720 by something in a fairly small cyclic group. 94 00:04:46,720 --> 00:04:48,530 And here the modeling is 8 isomorphic, 95 00:04:48,530 --> 00:04:52,060 so it preserves sums up to eight term sums. 96 00:04:55,060 --> 00:04:56,730 We had Bogolyubovs lemma so now we're 97 00:04:56,730 --> 00:04:59,310 inside a small cyclic group of a large subset 98 00:04:59,310 --> 00:05:01,200 of a small cyclic group. 99 00:05:01,200 --> 00:05:05,820 Then Bogolyubovs lemma says that 2a minus 2a contains 100 00:05:05,820 --> 00:05:10,080 a large bore set, of large structure within the situated 101 00:05:10,080 --> 00:05:11,210 subset. 102 00:05:11,210 --> 00:05:14,270 And last time we showed that the geometry of numbers, 103 00:05:14,270 --> 00:05:16,470 Minkowskis second term, one can deduce 104 00:05:16,470 --> 00:05:22,800 that every bore set of small dimension and small width 105 00:05:22,800 --> 00:05:24,310 contains-- 106 00:05:24,310 --> 00:05:31,980 of large width contains a proper GAP that's pretty large. 107 00:05:31,980 --> 00:05:34,650 So putting these two together, putting the last two things 108 00:05:34,650 --> 00:05:39,480 together, we obtain that if you have 109 00:05:39,480 --> 00:05:47,690 a subset of the cyclic group and n is prime-- 110 00:05:47,690 --> 00:05:52,670 OK, so here in previous statement n is prime. 111 00:05:52,670 --> 00:05:53,630 So n is prime. 112 00:05:56,591 --> 00:05:57,950 And a is pretty large. 113 00:06:00,680 --> 00:06:10,910 Then 2a minus 2a contains a proper generalized arithmetic 114 00:06:10,910 --> 00:06:18,560 progression of dimension at most alpha to the minus 2 115 00:06:18,560 --> 00:06:28,014 and size at least 1 over 40 to the d times n. 116 00:06:31,250 --> 00:06:33,510 So it's just starting from the size of a. 117 00:06:33,510 --> 00:06:36,390 2a minus 2a contains a pretty large GAP. 118 00:06:36,390 --> 00:06:38,640 So we're going to put all of these ingredients 119 00:06:38,640 --> 00:06:40,380 together and show that you can now 120 00:06:40,380 --> 00:06:46,580 contain the original set, a, in a small GAP. 121 00:06:46,580 --> 00:06:51,130 So just from knowing that some subset of it, 2a minus 2a, 122 00:06:51,130 --> 00:06:55,030 so think 2a prime minus 2a prime, contains this large GAP. 123 00:06:55,030 --> 00:06:57,030 We're going to use it to boost it up to a cover. 124 00:07:05,760 --> 00:07:08,160 So now let's prove Freiman's theorem. 125 00:07:13,940 --> 00:07:16,024 Using the modeling lemma-- 126 00:07:21,440 --> 00:07:24,740 using the modeling lemma-- the corollary of the modeling 127 00:07:24,740 --> 00:07:32,630 lemma-- we find that since a plus a is size 128 00:07:32,630 --> 00:07:35,760 at most k kind of size of a, there 129 00:07:35,760 --> 00:07:45,510 exists some prime n at most 2k to the 16 times a. 130 00:07:45,510 --> 00:07:51,050 And so I'm just copying the consequence of this modeling 131 00:07:51,050 --> 00:07:51,900 lemma. 132 00:07:51,900 --> 00:07:56,190 So I find a pretty large subset of a such that a prime is prime 133 00:07:56,190 --> 00:08:03,170 and 8 isomorphic to a subset of z mod n. 134 00:08:09,550 --> 00:08:17,690 Now, applying the final corollary 135 00:08:17,690 --> 00:08:29,220 with alpha being the size of this a prime, which 136 00:08:29,220 --> 00:08:38,049 is at least the size of a over n, which is at least 1 over 16 137 00:08:38,049 --> 00:08:43,179 to the times k to the power 16, so all constants. 138 00:08:43,179 --> 00:08:48,050 We see that 2a prime minus-- 139 00:08:48,050 --> 00:08:49,970 so let me actually-- let me change the letters 140 00:08:49,970 --> 00:08:54,380 and call a prime b so I don't have to keep on writing primes. 141 00:08:57,450 --> 00:09:02,090 So subset of a is called b. 142 00:09:02,090 --> 00:09:07,575 OK, so 2b minus 2b now contains a large GAP. 143 00:09:12,850 --> 00:09:18,800 And the GAP has dimension d bounded. 144 00:09:18,800 --> 00:09:22,370 So the dimension is bounded by alpha to the minus 2. 145 00:09:22,370 --> 00:09:23,755 So it's some constant. 146 00:09:28,880 --> 00:09:32,490 And the size is pretty large. 147 00:09:32,490 --> 00:09:39,775 So size is at least 1 over 40 d. 148 00:09:39,775 --> 00:09:41,150 If you only care about constants, 149 00:09:41,150 --> 00:09:44,900 just remember that everything that depends on k or d 150 00:09:44,900 --> 00:09:45,760 is a constant. 151 00:09:49,580 --> 00:09:50,080 OK. 152 00:09:53,920 --> 00:10:00,370 Because b is Freiman's 8 isomorphic, 153 00:10:00,370 --> 00:10:06,340 b is Freiman 8 isomorphic to-- 154 00:10:11,080 --> 00:10:13,450 ah, sorry. 155 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-- 156 00:10:23,800 --> 00:10:28,540 b is a subset of z mod n that a prime is 8 isomorphic, too. 157 00:10:28,540 --> 00:10:37,330 So since b is 8 isomorphic to a prime, every GAP in b-- 158 00:10:40,330 --> 00:10:45,360 so if you think about what 8 isomorphism preserves, 159 00:10:45,360 --> 00:10:53,880 you find that if you look at 2b minus 2b, it must be 2 prime-- 160 00:10:53,880 --> 00:11:00,760 2 isomorphic to 2a prime minus 2a prime. 161 00:11:00,760 --> 00:11:02,730 So the point of prime and isomorphism 162 00:11:02,730 --> 00:11:07,565 is that we just want to preserve enough additive structure. 163 00:11:07,565 --> 00:11:09,940 Well, we're doing to preserve all the additive structure, 164 00:11:09,940 --> 00:11:11,357 but just enough additive structure 165 00:11:11,357 --> 00:11:13,330 to do what we need to do. 166 00:11:13,330 --> 00:11:16,150 And being able to preserve an arithmetic progression, 167 00:11:16,150 --> 00:11:18,820 or in general or generalized arithmetic progression, 168 00:11:18,820 --> 00:11:23,400 requires you to preserve Freiman 2 isomorphism. 169 00:11:23,400 --> 00:11:25,280 And that's where the a comes in. 170 00:11:25,280 --> 00:11:28,180 So I want to analyze 2b minus 2b and I 171 00:11:28,180 --> 00:11:30,400 want that to preserve 2 isomorphisms. 172 00:11:30,400 --> 00:11:36,000 So initially I want b to preserve 8 isomorphisms. 173 00:11:36,000 --> 00:11:40,690 So 2b minus 2b is Freiman isomorphic to 2a prime minus 2a 174 00:11:40,690 --> 00:11:42,070 prime. 175 00:11:42,070 --> 00:11:50,690 So the GAP, which we found earlier in 2B minus 2B 176 00:11:50,690 --> 00:11:58,830 is mapped via this Freiman isomorphism 177 00:11:58,830 --> 00:12:04,880 to a proper GAP, which we'll call q, 178 00:12:04,880 --> 00:12:11,580 now setting aside 2a minus 2a and preserving 179 00:12:11,580 --> 00:12:15,456 the same dimension and size. 180 00:12:15,456 --> 00:12:19,200 So Freiman isomorphisms are good for preserving 181 00:12:19,200 --> 00:12:21,900 these partial additive structures like GAPs. 182 00:12:21,900 --> 00:12:22,610 Yes? 183 00:12:22,610 --> 00:12:25,060 AUDIENCE: So are we using this smaller structure to be 184 00:12:25,060 --> 00:12:26,040 [INAUDIBLE]? 185 00:12:33,357 --> 00:12:34,190 YUFEI ZHAO: Correct. 186 00:12:34,190 --> 00:12:37,430 So question is, we're using-- 187 00:12:37,430 --> 00:12:41,630 so because we have to pass the 2b minus 2b, 188 00:12:41,630 --> 00:12:44,600 we want 2b minus 2b to be prime and isomorphic to 2a 189 00:12:44,600 --> 00:12:45,710 prime minus 2a prime. 190 00:12:48,240 --> 00:12:52,730 So that's why in the proof I want b 191 00:12:52,730 --> 00:12:55,500 to be 8 isomorphic to a prime. 192 00:12:55,500 --> 00:12:58,800 So you see, so I'm skipping details of this step. 193 00:12:58,800 --> 00:13:02,330 But if you read the definition of Freiman's s isomorphism, 194 00:13:02,330 --> 00:13:04,111 you see that this implication holds. 195 00:13:04,111 --> 00:13:06,465 AUDIENCE: [INAUDIBLE] 196 00:13:06,465 --> 00:13:07,090 YUFEI ZHAO: No. 197 00:13:07,090 --> 00:13:10,367 2 isomorphism is a weaker condition. 198 00:13:13,110 --> 00:13:17,170 2 isomorphism just means that you are preserving two y sums. 199 00:13:21,030 --> 00:13:25,870 So think about the definition of Freiman 2 isomorphisms. 200 00:13:25,870 --> 00:13:30,160 In particular, if two sets are Freiman 2 isomorphic 201 00:13:30,160 --> 00:13:33,160 and you have a arithmetic progression in one, 202 00:13:33,160 --> 00:13:35,620 then that arithmetic progression is also an arithmetic 203 00:13:35,620 --> 00:13:38,800 progression in the other. 204 00:13:38,800 --> 00:13:41,310 So it's just enough additive structure 205 00:13:41,310 --> 00:13:43,860 to preserve things like arithmetic progressions 206 00:13:43,860 --> 00:13:48,456 and generalized arithmetic progressions. 207 00:13:48,456 --> 00:13:49,360 OK. 208 00:13:49,360 --> 00:13:56,020 So we found this large GAP in 2a minus 2a. 209 00:13:56,020 --> 00:13:58,180 So this is very good. 210 00:13:58,180 --> 00:14:01,540 So we wanted to contain a in the GAP. 211 00:14:01,540 --> 00:14:03,800 Seems that by now we're doing something slightly 212 00:14:03,800 --> 00:14:04,900 in the opposite direction. 213 00:14:04,900 --> 00:14:10,917 We find a large GAP within 2a minus 2a. 214 00:14:10,917 --> 00:14:12,500 But something we've seen before, we're 215 00:14:12,500 --> 00:14:16,100 going to use this to boost ourselves to a covering of a 216 00:14:16,100 --> 00:14:18,750 via the Ruzsa covering lemma. 217 00:14:18,750 --> 00:14:21,540 So once you find this large structure, 218 00:14:21,540 --> 00:14:27,800 you can now try to take translates of it to cover a. 219 00:14:27,800 --> 00:14:30,170 And this is-- if there's any takeaway from the spirit 220 00:14:30,170 --> 00:14:31,970 of this proof is this idea. 221 00:14:31,970 --> 00:14:34,160 Even though I want to cover the whole set, it's OK. 222 00:14:34,160 --> 00:14:37,220 I just find a large structure within it 223 00:14:37,220 --> 00:14:39,140 and then I use translaters to cover. 224 00:14:41,770 --> 00:14:43,450 How do we do this? 225 00:14:43,450 --> 00:14:50,350 So since q is containing 2a minus 2a, 226 00:14:50,350 --> 00:14:53,650 we find that q plus a is containing 3a minus 2a. 227 00:15:00,120 --> 00:15:07,110 Therefore, by Plünnecke-Ruzsa-- by Plünnecke-Ruzsa inequality, 228 00:15:07,110 --> 00:15:13,790 the size of cube plus a is at most the size of 3a minus 2a, 229 00:15:13,790 --> 00:15:17,720 which is, at most, k to the fifth power times the size 230 00:15:17,720 --> 00:15:18,220 of a. 231 00:15:23,120 --> 00:15:27,060 And I claim that this final quantity is also not 232 00:15:27,060 --> 00:15:34,120 so different from the size of cube, because-- 233 00:15:34,120 --> 00:15:35,590 so all of these-- 234 00:15:35,590 --> 00:15:37,672 I mean, the point here is, we are 235 00:15:37,672 --> 00:15:39,130 doing all of these transformations, 236 00:15:39,130 --> 00:15:41,830 passing down to subsets, putting something bigger, putting-- 237 00:15:41,830 --> 00:15:44,260 getting to something smaller, but each time we only 238 00:15:44,260 --> 00:15:47,910 loose something that is polynomial in k. 239 00:15:47,910 --> 00:15:49,700 We're not losing much more. 240 00:15:49,700 --> 00:15:52,450 I am also only losing a constant factor. 241 00:15:52,450 --> 00:15:56,110 There is sometimes a bit more than polynomial, 242 00:15:56,110 --> 00:16:01,280 but any case, we're losing only a constant factor at each step. 243 00:16:01,280 --> 00:16:12,110 So in particular, since n upper bounds the size of a prime 244 00:16:12,110 --> 00:16:17,610 is here where we ended up embedding into z modern, 245 00:16:17,610 --> 00:16:23,270 n is larger than a prime, which is at least a constant fraction 246 00:16:23,270 --> 00:16:31,390 of a and the size of q is at least 1 247 00:16:31,390 --> 00:16:38,210 over 40 raised d times n. 248 00:16:38,210 --> 00:16:43,870 So we find that this bound-- 249 00:16:43,870 --> 00:16:46,980 upper bound earlier on q plus a. 250 00:16:46,980 --> 00:16:56,360 We can write it in terms of size of cube 251 00:16:56,360 --> 00:17:01,760 where k prime is-- you put all of these numbers together. 252 00:17:01,760 --> 00:17:05,180 What it is specifically doesn't matter so much, 253 00:17:05,180 --> 00:17:06,589 other than that it is a constant. 254 00:17:11,829 --> 00:17:14,359 d is polynomial in k. 255 00:17:14,359 --> 00:17:18,050 So what we have here is something that is exponential, 256 00:17:18,050 --> 00:17:19,069 polynomial of k. 257 00:17:27,790 --> 00:17:31,110 OK, so now we're in a position to apply the Ruzsa covering 258 00:17:31,110 --> 00:17:32,400 lemma. 259 00:17:32,400 --> 00:17:35,440 So look at that statement up there. 260 00:17:35,440 --> 00:17:37,210 So what is the saying, that a plus q 261 00:17:37,210 --> 00:17:41,310 looks like it could be covered by q, just in terms of size. 262 00:17:41,310 --> 00:17:45,450 So I should expect to cover a by a small number of translates 263 00:17:45,450 --> 00:17:46,590 of q minus q. 264 00:17:51,430 --> 00:17:57,690 So by covering lemma, a is containing some x plus q 265 00:17:57,690 --> 00:18:05,630 minus q for some x in a where the size of x 266 00:18:05,630 --> 00:18:07,640 is at most k prime. 267 00:18:11,350 --> 00:18:14,920 I claim-- so we've covered a by something. 268 00:18:14,920 --> 00:18:18,740 q is a GAP. 269 00:18:18,740 --> 00:18:22,690 x is a bounded size set, and I claim 270 00:18:22,690 --> 00:18:27,950 that this is the type of object that we're happy to have. 271 00:18:27,950 --> 00:18:31,980 Just to spell out some details, first 272 00:18:31,980 --> 00:18:45,480 note that x is contained in a GAP dimension x or x minus 1 273 00:18:45,480 --> 00:18:48,217 with length 2 in each direction. 274 00:18:54,560 --> 00:18:56,820 So add a new direction for every element of x. 275 00:18:56,820 --> 00:19:00,450 It's wasteful, but everything's constant. 276 00:19:00,450 --> 00:19:07,230 And recall that the dimension of Q as a GAP is d. 277 00:19:07,230 --> 00:19:22,980 So x plus q minus q is contained in a GAP of dimension. 278 00:19:22,980 --> 00:19:24,900 OK, so what's the dimension? 279 00:19:24,900 --> 00:19:28,890 So when I do q minus q, it's like taking a box 280 00:19:28,890 --> 00:19:32,490 and doubling its dimension-- doubling its lengths. 281 00:19:32,490 --> 00:19:34,870 I'm not changing the number of dimensions. 282 00:19:34,870 --> 00:19:37,530 So the dimension of q minus q is still d. 283 00:19:37,530 --> 00:19:41,250 The dimension of x is, at most, the size of x. 284 00:19:44,220 --> 00:19:46,850 All of these things are constants. 285 00:19:46,850 --> 00:19:47,755 So we're happy. 286 00:19:47,755 --> 00:19:52,210 But to spell it out, the constant here is-- 287 00:19:52,210 --> 00:19:55,800 well, k prime is what we wrote up there. 288 00:19:59,800 --> 00:20:04,040 So this is a constant. 289 00:20:04,040 --> 00:20:11,570 And the size-- so what is the size of the GAP 290 00:20:11,570 --> 00:20:13,940 that contains this guy here? 291 00:20:13,940 --> 00:20:19,730 So I'm expanding x to a GAP by adding a new direction 292 00:20:19,730 --> 00:20:22,340 for every element of x. 293 00:20:22,340 --> 00:20:25,250 And I might expand that size a little bit. 294 00:20:25,250 --> 00:20:29,930 But the size of this GAP that contains x is no more than 2 295 00:20:29,930 --> 00:20:33,100 to the power x-- 296 00:20:33,100 --> 00:20:37,220 2 raised to the size of x. 297 00:20:37,220 --> 00:20:42,860 What is the size of GAP q minus q? 298 00:20:42,860 --> 00:20:45,740 So q is the GAP of dimension d. 299 00:20:45,740 --> 00:20:48,050 And we know that a GAP of dimension d 300 00:20:48,050 --> 00:20:52,710 has doubling constant and those 2 to the d-- 301 00:20:52,710 --> 00:20:56,960 2 to the d times the size of q. 302 00:20:56,960 --> 00:20:58,060 OK. 303 00:20:58,060 --> 00:21:03,580 And because q is contained in qa minus 2a, 304 00:21:03,580 --> 00:21:10,340 we find that q is contained in 2a minus 2a. 305 00:21:10,340 --> 00:21:12,830 And 2 to the x, well, I know what the site of x 306 00:21:12,830 --> 00:21:22,800 is bounded by, it's k prime plus the size of x is-- 307 00:21:22,800 --> 00:21:24,390 size of x is, at most, k prime. 308 00:21:24,390 --> 00:21:27,890 And then I have 2 to the d over here, 309 00:21:27,890 --> 00:21:34,150 so 2a minus 2a by Plünnecke-Ruzsa is at most k 310 00:21:34,150 --> 00:21:36,490 to the 4 times the size of a. 311 00:21:36,490 --> 00:21:39,820 OK, you put everything together, we find that this bound here 312 00:21:39,820 --> 00:21:44,790 is doubly exponential in the-- it's a polynomial in k. 313 00:21:47,920 --> 00:21:50,490 And that's it. 314 00:21:50,490 --> 00:21:52,240 This proves Freiman's theory. 315 00:21:58,792 --> 00:22:03,785 Now, to recap-- we went through several steps. 316 00:22:03,785 --> 00:22:07,050 So first, using the modeling lemma, 317 00:22:07,050 --> 00:22:09,990 we know that if a set a has small doubling, 318 00:22:09,990 --> 00:22:14,940 then we can pass a large part of a to a relatively 319 00:22:14,940 --> 00:22:17,290 small cyclic group. 320 00:22:17,290 --> 00:22:20,710 Going to work inside our cyclic group. 321 00:22:20,710 --> 00:22:28,220 Using Bogolyubovs lemma and its geometry of numbers corollary, 322 00:22:28,220 --> 00:22:30,890 we find that inside the cyclic group, 323 00:22:30,890 --> 00:22:33,950 the corresponding set, which we called b, 324 00:22:33,950 --> 00:22:37,330 is such that 2b minus 2b contains a large GAP. 325 00:22:40,390 --> 00:22:43,450 We pass that GAP back to the original set 326 00:22:43,450 --> 00:22:49,290 a because we are preserving 8 isomorphisms Freiman 8 327 00:22:49,290 --> 00:22:51,260 isomorphisms in Ruzsa modeling lemma 328 00:22:51,260 --> 00:22:53,600 so we can pass to the original set a 329 00:22:53,600 --> 00:22:58,770 and find the large GAP in 2a minus 2a. 330 00:22:58,770 --> 00:23:02,880 Once we find this large GAP 2a minus 2a, 331 00:23:02,880 --> 00:23:06,510 then we're going to use the Ruzsa covering 332 00:23:06,510 --> 00:23:15,390 lemma to contain a inside a small number of translates 333 00:23:15,390 --> 00:23:16,560 of this GAP. 334 00:23:20,060 --> 00:23:20,750 OK. 335 00:23:20,750 --> 00:23:23,450 You put all of these things together and the appropriate 336 00:23:23,450 --> 00:23:27,310 bounds coming from Plünnecke-Ruzsa inequalities 337 00:23:27,310 --> 00:23:30,218 and you get a final theorem. 338 00:23:30,218 --> 00:23:32,010 And this is the proof of Freiman's theorem. 339 00:23:34,735 --> 00:23:36,185 AUDIENCE: [INAUDIBLE] 340 00:23:36,185 --> 00:23:36,810 YUFEI ZHAO: OK. 341 00:23:36,810 --> 00:23:40,420 The question is how do you make it proper? 342 00:23:40,420 --> 00:23:44,320 Up until the step with q, it is still proper. 343 00:23:44,320 --> 00:23:48,790 So the very last step over here it is-- 344 00:23:48,790 --> 00:23:51,610 you might have destroyed properness. 345 00:23:51,610 --> 00:23:54,410 So this proof here doesn't give you properness. 346 00:23:54,410 --> 00:23:58,000 So I mentioned at the beginning that in Freiman's theorem, 347 00:23:58,000 --> 00:24:02,170 you can obtain properness of the additional arguments. 348 00:24:02,170 --> 00:24:03,660 So that I'm not going to show. 349 00:24:03,660 --> 00:24:08,125 There's some more work which is related to geometry of numbers. 350 00:24:10,940 --> 00:24:13,825 So for example, you can look up in the textbook by Tao and Vu, 351 00:24:13,825 --> 00:24:21,340 and see how to get from a GAP to contain it in a proper GAP, 352 00:24:21,340 --> 00:24:25,780 without losing too much in terms of size. 353 00:24:25,780 --> 00:24:28,532 So think about it this way-- 354 00:24:28,532 --> 00:24:30,490 when do you have something which is not proper? 355 00:24:30,490 --> 00:24:33,160 When you have-- and if some linear dependence, 356 00:24:33,160 --> 00:24:35,680 you have some integer linear dependence. 357 00:24:35,680 --> 00:24:40,060 And in that case, you kind of lost a dimension. 358 00:24:40,060 --> 00:24:42,940 When you have improperness, you actually go down a dimension. 359 00:24:42,940 --> 00:24:46,270 But then you need to salvage the size, 360 00:24:46,270 --> 00:24:48,830 make sure that the size doesn't blow up too much. 361 00:24:48,830 --> 00:24:50,960 And so there are some arguments to be done there. 362 00:24:50,960 --> 00:24:52,810 And we're not going to do it here. 363 00:24:52,810 --> 00:24:54,960 AUDIENCE: Well, I guess my [INAUDIBLE] 364 00:24:54,960 --> 00:24:58,880 do they change [INAUDIBLE] within the proof, 365 00:24:58,880 --> 00:25:01,510 like say [INAUDIBLE] q or whatever? 366 00:25:01,510 --> 00:25:05,060 Or do they use the same proof, but later on, 367 00:25:05,060 --> 00:25:08,725 say that [INAUDIBLE]? 368 00:25:08,725 --> 00:25:09,600 YUFEI ZHAO: OK, good. 369 00:25:09,600 --> 00:25:11,340 Yeah, so the question is, to get properness, 370 00:25:11,340 --> 00:25:13,882 do I have to modify the proof, or can I use Freiman's theorem 371 00:25:13,882 --> 00:25:15,350 as witness of a black box. 372 00:25:15,350 --> 00:25:17,020 So my understanding is that I can 373 00:25:17,020 --> 00:25:20,422 use the statement as a black box and obtain properness. 374 00:25:20,422 --> 00:25:21,880 But if you want to get good bounds, 375 00:25:21,880 --> 00:25:24,250 maybe you have to go into the proof, although that, 376 00:25:24,250 --> 00:25:24,820 I'm not sure. 377 00:25:27,450 --> 00:25:31,500 OK, any more questions? 378 00:25:31,500 --> 00:25:34,670 So this took a while. 379 00:25:34,670 --> 00:25:38,620 This was the most involved proof we've done in this course 380 00:25:38,620 --> 00:25:40,630 so far, in proving Freiman's theorem. 381 00:25:40,630 --> 00:25:43,500 We had to develop a large number of tools. 382 00:25:43,500 --> 00:25:44,930 And we came up-- 383 00:25:44,930 --> 00:25:47,140 so we eventually arrived at-- 384 00:25:47,140 --> 00:25:48,260 it's a beautiful theorem. 385 00:25:48,260 --> 00:25:51,190 So this is a fantastic result that 386 00:25:51,190 --> 00:25:55,030 gives you an inverse structure, something-- 387 00:25:55,030 --> 00:25:57,520 we know that GAP's have small doubling. 388 00:25:57,520 --> 00:26:00,170 And conversely, if something has a small doubling, 389 00:26:00,170 --> 00:26:03,650 it has to, in some sense, look like a GAP. 390 00:26:03,650 --> 00:26:05,590 So you see that the proof is quite involved 391 00:26:05,590 --> 00:26:08,780 and has a lot of beautiful ideas. 392 00:26:08,780 --> 00:26:10,690 In the remainder of today's lecture, 393 00:26:10,690 --> 00:26:15,250 I want to present some remarks on additional extensions 394 00:26:15,250 --> 00:26:17,710 and generalizations of Freiman's theorem. 395 00:26:17,710 --> 00:26:20,660 And while we're not going to do any proofs, 396 00:26:20,660 --> 00:26:23,230 there's a lot of deep and beautiful mathematics 397 00:26:23,230 --> 00:26:26,230 that are involved in the subject. 398 00:26:26,230 --> 00:26:31,150 So I want to take you on a tour through some more things 399 00:26:31,150 --> 00:26:34,480 that we can talk about when it comes to Freiman's theorem. 400 00:26:34,480 --> 00:26:38,350 But first, let me mention a few things 401 00:26:38,350 --> 00:26:42,040 that I mentioned very quickly when we first 402 00:26:42,040 --> 00:26:43,990 introduced Freiman's theorem, namely 403 00:26:43,990 --> 00:26:45,310 some remarks on the bounds. 404 00:26:52,120 --> 00:26:55,890 So the proof that we just saw gives you 405 00:26:55,890 --> 00:27:00,550 a bound, which is basically exponential in the dimension 406 00:27:00,550 --> 00:27:04,770 and doubly exponential for the size blow-up. 407 00:27:04,770 --> 00:27:08,530 They're all constants, so if you only care about constants, 408 00:27:08,530 --> 00:27:10,540 then this is just fine. 409 00:27:10,540 --> 00:27:13,790 But you may ask, are we losing too much here? 410 00:27:13,790 --> 00:27:18,120 What is the right type of dependence? 411 00:27:18,120 --> 00:27:20,610 So what is the right type of dependence? 412 00:27:20,610 --> 00:27:23,030 So we saw an example. 413 00:27:23,030 --> 00:27:28,020 So we saw an example where if you start with A being 414 00:27:28,020 --> 00:27:35,930 a highly dissociated set, where there is basically 415 00:27:35,930 --> 00:27:40,370 no additive structure within A, then you do need-- 416 00:27:40,370 --> 00:27:49,600 so this example shows that you cannot do better than 417 00:27:49,600 --> 00:27:50,900 polynomial-- 418 00:27:50,900 --> 00:27:58,080 well, actually, than linear in K, in the dimension, 419 00:27:58,080 --> 00:28:03,240 and exponential in the size blow-up. 420 00:28:03,240 --> 00:28:06,030 So in particular, you do need to blow up 421 00:28:06,030 --> 00:28:11,690 the size by some exponential quantity in K. 422 00:28:11,690 --> 00:28:16,450 So here, K is roughly the size of A over 2 in this example. 423 00:28:16,450 --> 00:28:19,130 And you can create modifications of the example 424 00:28:19,130 --> 00:28:22,020 to keep K constant and A getting larger. 425 00:28:22,020 --> 00:28:25,160 But the point is that you cannot do better than this type 426 00:28:25,160 --> 00:28:27,740 of dependence simply from that example. 427 00:28:27,740 --> 00:28:31,100 And it's conjecture that that is the truth. 428 00:28:31,100 --> 00:28:33,590 We're almost there in proving this conjecture, 429 00:28:33,590 --> 00:28:36,020 but not quite, although the proof that we just gave 430 00:28:36,020 --> 00:28:41,400 is somewhat far, because you lose an exponent in each bound. 431 00:28:41,400 --> 00:28:45,210 There is a refinement of the final step in the argument, 432 00:28:45,210 --> 00:28:46,680 so let me comment on that. 433 00:28:46,680 --> 00:28:53,750 So we can refine the final step or the final steps 434 00:28:53,750 --> 00:28:59,045 in the proof to get polynomial bounds. 435 00:29:04,000 --> 00:29:10,690 And to get a polynomial bounds, which 436 00:29:10,690 --> 00:29:13,570 is much more in the right ballpark 437 00:29:13,570 --> 00:29:15,710 compared to what we got. 438 00:29:15,710 --> 00:29:18,820 And the idea is basically over here, 439 00:29:18,820 --> 00:29:20,380 we used the Ruzsa covering lemma. 440 00:29:20,380 --> 00:29:23,500 So we started with that Q up there. 441 00:29:23,500 --> 00:29:30,660 So up until this point, you should think of this step 442 00:29:30,660 --> 00:29:34,390 as everything coming from Bogolyubov and its corollary. 443 00:29:34,390 --> 00:29:36,650 So that stays the same. 444 00:29:36,650 --> 00:29:39,830 And now the question is starting with our Q, what would you use? 445 00:29:39,830 --> 00:29:45,340 How would you use this Q to try to cover it? 446 00:29:45,340 --> 00:29:48,673 Well, what we do, we apply Ruzsa covering lemma. 447 00:29:48,673 --> 00:29:50,840 Remember how the proof of Ruzsa covering lemma goes. 448 00:29:50,840 --> 00:29:54,920 You take a maximal set of translates, 449 00:29:54,920 --> 00:29:56,260 disjoint translates. 450 00:29:56,260 --> 00:29:58,010 And if you blow everything up a factor 2, 451 00:29:58,010 --> 00:30:00,500 then you've got a cover. 452 00:30:00,500 --> 00:30:03,030 But it turns out to be somewhat wasteful. 453 00:30:03,030 --> 00:30:06,860 And you see, there was a lot of waste in going from x to 2 454 00:30:06,860 --> 00:30:08,760 to the x. 455 00:30:08,760 --> 00:30:12,550 So you could do that step more slowly. 456 00:30:12,550 --> 00:30:21,100 So starting with Q, cover now some, not all 457 00:30:21,100 --> 00:30:34,500 of A. So cover parts of A by translates of 2 minus Q, say. 458 00:30:34,500 --> 00:30:36,390 So we do Ruzsa covering lemma, you 459 00:30:36,390 --> 00:30:40,110 don't cover the whole thing, but nibble away, cover 460 00:30:40,110 --> 00:30:44,870 a little bit, and then look at the thing that you get, 461 00:30:44,870 --> 00:30:51,230 which is that Q will become some new thing, let's say Q1. 462 00:30:51,230 --> 00:31:00,530 And now cover more by Q1 minus Q1. 463 00:31:00,530 --> 00:31:06,670 So apparently, if you do the covering step more slowly, 464 00:31:06,670 --> 00:31:08,530 you can obtain better bounds. 465 00:31:08,530 --> 00:31:14,320 And that's enough to save you this exponent, 466 00:31:14,320 --> 00:31:19,230 to go down to polynomial-type bounds for Freiman's theorem. 467 00:31:19,230 --> 00:31:21,830 So I'm not giving details, but this is roughly the idea. 468 00:31:21,830 --> 00:31:26,250 So you can modify the final step to obtain this bound. 469 00:31:26,250 --> 00:31:35,410 The best bound so far is due to Tom Sanders, who 470 00:31:35,410 --> 00:31:40,300 proved Freiman's theorem for bounds 471 00:31:40,300 --> 00:31:46,540 on dimension that's like K times poly log K, 472 00:31:46,540 --> 00:31:54,310 and the size blowup to be E to the K times poly log K. 473 00:31:54,310 --> 00:31:57,310 So in other words, other than this polylogarithmic factor, 474 00:31:57,310 --> 00:32:00,740 it's basically the right answer. 475 00:32:00,740 --> 00:32:04,220 And so this proof is much more sophisticated. 476 00:32:04,220 --> 00:32:07,390 So it goes much more in depth into analyzing 477 00:32:07,390 --> 00:32:09,530 the structure of set addition. 478 00:32:09,530 --> 00:32:12,250 So Sanders has a very nice survey article 479 00:32:12,250 --> 00:32:14,500 called "The structure of Set Addition" 480 00:32:14,500 --> 00:32:19,090 that analyzes some of the modern techniques that 481 00:32:19,090 --> 00:32:21,180 are used to prove these types of results. 482 00:32:28,480 --> 00:32:30,220 There is one more issue, which I want 483 00:32:30,220 --> 00:32:33,360 to discuss at length in the second half of this lecture, 484 00:32:33,360 --> 00:32:36,940 which is that you might be very unhappy 485 00:32:36,940 --> 00:32:41,770 with this exponential blowup, because if you think about what 486 00:32:41,770 --> 00:32:43,720 happens in these examples-- 487 00:32:43,720 --> 00:32:46,440 I mean, not the examples, but if you think about what happens, 488 00:32:46,440 --> 00:32:49,390 like the spirit of what we're trying to say, 489 00:32:49,390 --> 00:32:53,390 Freiman's theorem is some kind of an inverse theorem. 490 00:32:53,390 --> 00:32:57,410 And to go forward, you're trying to say 491 00:32:57,410 --> 00:33:05,060 that if you have a GAP of dimension d, then 492 00:33:05,060 --> 00:33:11,000 the size blowup is like 2 to the d. 493 00:33:11,000 --> 00:33:22,130 So we want to say some structure applies small doubling, 494 00:33:22,130 --> 00:33:25,200 and Freiman's theorem tells the reverse, that you 495 00:33:25,200 --> 00:33:33,030 have small doubling, then you obtain this structure. 496 00:33:33,030 --> 00:33:39,420 And seems like you are losing. 497 00:33:39,420 --> 00:33:44,340 Getting from here to here, there is a polynomial type of loss, 498 00:33:44,340 --> 00:33:46,290 whereas going from here to here, it 499 00:33:46,290 --> 00:33:49,950 seems that we're incurring some exponential type of loss. 500 00:33:49,950 --> 00:33:52,740 And it'll be nice to have some kind of inverse theorem 501 00:33:52,740 --> 00:33:57,180 that also preserves these relationships qualitatively. 502 00:33:57,180 --> 00:33:59,070 So that may not make sense in this moment, 503 00:33:59,070 --> 00:34:00,903 but we'll get back to it later this lecture. 504 00:34:04,250 --> 00:34:05,660 Point is, there's more, much more 505 00:34:05,660 --> 00:34:07,520 to be said about the bounds here, 506 00:34:07,520 --> 00:34:09,530 even though right now it looks as if they're 507 00:34:09,530 --> 00:34:12,860 very close to each other. 508 00:34:12,860 --> 00:34:15,980 One more thing that I want to expand on 509 00:34:15,980 --> 00:34:19,300 is, we've stated and proved Freiman's theorem 510 00:34:19,300 --> 00:34:21,409 in the integers. 511 00:34:21,409 --> 00:34:26,850 And you might ask, what about in other groups? 512 00:34:26,850 --> 00:34:31,620 We also proved Freiman's theorem in F2 to the m, or more 513 00:34:31,620 --> 00:34:33,989 generally, groups of bounded exponent 514 00:34:33,989 --> 00:34:39,270 or bounded portion, so abelian groups of bounded exponent. 515 00:34:39,270 --> 00:34:46,600 For general abelian groups, so Freiman's theorem 516 00:34:46,600 --> 00:34:54,900 in general abelian groups, you might ask what happens here? 517 00:34:54,900 --> 00:34:58,230 And in some sense what is even the statement of the theorem? 518 00:34:58,230 --> 00:35:01,120 So we want something which combines, 519 00:35:01,120 --> 00:35:04,280 somehow, two different types of behavior. 520 00:35:04,280 --> 00:35:07,670 On one hand, you have z, which is what we just did. 521 00:35:07,670 --> 00:35:10,810 And here the model structures are GAP's. 522 00:35:10,810 --> 00:35:14,330 And on the other hand, we have, which we also proved, 523 00:35:14,330 --> 00:35:17,570 things like F2 to the m, where the model structures are 524 00:35:17,570 --> 00:35:18,200 subgroups. 525 00:35:22,520 --> 00:35:24,270 And there's a sense in which these are not 526 00:35:24,270 --> 00:35:26,400 the GAP's and subgroups. 527 00:35:26,400 --> 00:35:27,900 They have some similar properties, 528 00:35:27,900 --> 00:35:31,350 but they're not really like each other. 529 00:35:31,350 --> 00:35:33,500 So now if I give you a general group, which 530 00:35:33,500 --> 00:35:36,740 might be some combination of infinite torsion 531 00:35:36,740 --> 00:35:39,590 or very large torsion elements versus very small torsion 532 00:35:39,590 --> 00:35:42,410 elements-- so for example, take a Cartesian product 533 00:35:42,410 --> 00:35:43,820 of these groups. 534 00:35:43,820 --> 00:35:45,900 Is there a Freiman's theorem? 535 00:35:45,900 --> 00:35:48,780 And what does such a theorem look like? 536 00:35:48,780 --> 00:35:51,340 What are the structures? 537 00:35:51,340 --> 00:35:54,080 What are the subsets of bounded doubling? 538 00:35:58,040 --> 00:36:01,590 So that's kind of the thing we want to think about. 539 00:36:01,590 --> 00:36:05,010 So it turns out for Freiman's theorem in general abelian 540 00:36:05,010 --> 00:36:06,330 groups-- 541 00:36:06,330 --> 00:36:07,570 so there is a theorem. 542 00:36:07,570 --> 00:36:10,950 So this theorem was proved by Green and Ruzsa. 543 00:36:14,880 --> 00:36:17,730 So following a very similar type of proof framework, 544 00:36:17,730 --> 00:36:21,330 although the individual steps, in particular 545 00:36:21,330 --> 00:36:24,840 the modeling lemma needs to be modified. 546 00:36:24,840 --> 00:36:27,180 And let me tell you what the statement is. 547 00:36:27,180 --> 00:36:32,200 So the common generalization of GAP's and subgroups is 548 00:36:32,200 --> 00:36:34,740 something called a "co-set progression." 549 00:36:41,330 --> 00:36:46,790 So a co-set progression is a subset 550 00:36:46,790 --> 00:36:54,070 which is a direct sum of the form P plus H, 551 00:36:54,070 --> 00:36:57,180 where P is a proper GAP. 552 00:36:59,845 --> 00:37:03,990 So the definition of GAP works just fine in every abelian 553 00:37:03,990 --> 00:37:04,490 group. 554 00:37:04,490 --> 00:37:08,390 You start with the initial point, a few directions, 555 00:37:08,390 --> 00:37:13,590 and you look at a grid expansion of those directions. 556 00:37:13,590 --> 00:37:17,120 P is a proper GAP, and H is a subgroup. 557 00:37:20,300 --> 00:37:26,100 And here, the direct sum refers to the fact that every-- 558 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 559 00:37:34,330 --> 00:37:39,310 prime in the set P, and H and H prime in the set H, 560 00:37:39,310 --> 00:37:43,780 then P equals to P prime and H equals to H prime. 561 00:37:43,780 --> 00:37:46,150 So every element in here is written 562 00:37:46,150 --> 00:37:49,872 in a unique way as some P plus some H. So 563 00:37:49,872 --> 00:37:51,330 that's what I mean by "direct sum." 564 00:37:54,400 --> 00:37:58,930 For such an object, so such a co-set progression, 565 00:37:58,930 --> 00:38:09,640 I call its dimension to be the dimension of the GAP, P. 566 00:38:09,640 --> 00:38:12,820 And its size in this case, actually, 567 00:38:12,820 --> 00:38:16,930 is just the size of the set, which is also the size of P 568 00:38:16,930 --> 00:38:23,620 times the size of H. So the theorem 569 00:38:23,620 --> 00:38:36,960 is that if A is a subset of an arbitrary abelian group 570 00:38:36,960 --> 00:38:49,650 and it has bounded doubling, then A is contained 571 00:38:49,650 --> 00:39:03,930 in a co-set progression of bounded dimension and size, 572 00:39:03,930 --> 00:39:10,596 bounded blowup of the size of A. 573 00:39:10,596 --> 00:39:14,400 And here, these constants D and K are universal. 574 00:39:14,400 --> 00:39:17,360 They do not depend on the group. 575 00:39:17,360 --> 00:39:19,050 So there are some specific numbers, 576 00:39:19,050 --> 00:39:20,480 functions you can write down. 577 00:39:20,480 --> 00:39:24,040 They do not depend on the group. 578 00:39:24,040 --> 00:39:29,600 So this theorem gives you the characterization 579 00:39:29,600 --> 00:39:32,668 of subsets in general abelian groups 580 00:39:32,668 --> 00:39:33,710 that have small doubling. 581 00:39:36,610 --> 00:39:38,056 Any questions? 582 00:39:38,056 --> 00:39:39,808 Yes? 583 00:39:39,808 --> 00:39:40,690 AUDIENCE: [INAUDIBLE] 584 00:39:40,690 --> 00:39:41,510 YUFEI ZHAO: That's a good question. 585 00:39:41,510 --> 00:39:43,177 So I think you could go into their paper 586 00:39:43,177 --> 00:39:48,050 and see that you can get polynomial type bounds. 587 00:39:48,050 --> 00:39:51,920 And I think Sander's results also 588 00:39:51,920 --> 00:39:57,020 work for this type of setting to give you these type of bounds. 589 00:39:57,020 --> 00:39:59,960 But I-- yes, so you should look into Sanders' paper, 590 00:39:59,960 --> 00:40:00,830 and he will explain. 591 00:40:00,830 --> 00:40:04,200 I think in Sanders' paper he walks in general abelian 592 00:40:04,200 --> 00:40:04,700 groups. 593 00:40:07,960 --> 00:40:10,640 The next question I want to address is-- 594 00:40:10,640 --> 00:40:15,520 well, what do you think is the next question? 595 00:40:15,520 --> 00:40:20,370 Non-abelian groups, so Freiman's theorem in non-abelian groups, 596 00:40:20,370 --> 00:40:25,250 or rather the Freiman problem in non-abelian groups. 597 00:40:32,500 --> 00:40:35,710 So here's a basic question-- if I give you 598 00:40:35,710 --> 00:40:41,390 a non-abelian group, what subsets have bounded doubling? 599 00:40:41,390 --> 00:40:43,730 Of course, the examples from abelian groups 600 00:40:43,730 --> 00:40:47,330 also work in non-abelian groups, where you have subgroups, 601 00:40:47,330 --> 00:40:50,930 you have generalized arithmetical progressions. 602 00:40:50,930 --> 00:40:54,290 But are there genuinely new examples 603 00:40:54,290 --> 00:40:57,980 of sets in non-abelian groups that have bounded doubling? 604 00:41:01,160 --> 00:41:03,370 So think about that, and let's take a quick break. 605 00:41:07,790 --> 00:41:11,420 Can you think of examples in non-abelian groups 606 00:41:11,420 --> 00:41:14,518 that have small doubling, that do not come from the examples 607 00:41:14,518 --> 00:41:15,560 that we have seen before? 608 00:41:22,470 --> 00:41:24,090 So let me show you one construction. 609 00:41:24,090 --> 00:41:26,690 And this is that important construction 610 00:41:26,690 --> 00:41:27,740 for non-abelian groups. 611 00:41:34,662 --> 00:41:35,370 So it has a name. 612 00:41:35,370 --> 00:41:43,690 It's called a discrete Heisenberg group, 613 00:41:43,690 --> 00:41:51,820 which is the matrix group consisting of matrices that 614 00:41:51,820 --> 00:41:54,130 look like what I've written. 615 00:41:54,130 --> 00:41:56,740 So you have integer entries above the diagonal, 616 00:41:56,740 --> 00:42:01,090 1 on the diagonal, and 0 below the diagonal. 617 00:42:01,090 --> 00:42:03,030 So let's do some elementary matrix 618 00:42:03,030 --> 00:42:08,680 multiplication to see how group multiplication in this group 619 00:42:08,680 --> 00:42:10,190 works. 620 00:42:10,190 --> 00:42:17,910 So if I have two such matrices, I multiply them together. 621 00:42:17,910 --> 00:42:20,475 And then you see that the diagonal 622 00:42:20,475 --> 00:42:22,240 is preserved, of course. 623 00:42:22,240 --> 00:42:28,800 But this entry over here is simply addition. 624 00:42:28,800 --> 00:42:31,860 So this entry here is just addition. 625 00:42:31,860 --> 00:42:34,140 This entry over here is also addition. 626 00:42:37,040 --> 00:42:40,870 And the top right entry is a bit more complicated. 627 00:42:40,870 --> 00:42:45,100 It's some addition, but there's an additional twist. 628 00:42:52,230 --> 00:42:53,900 So this is how matrix multiplication 629 00:42:53,900 --> 00:42:55,642 works in this group. 630 00:42:55,642 --> 00:42:57,350 I mean, this is how matrix multiplication 631 00:42:57,350 --> 00:43:00,530 works, but in terms of elements of this group, that's 632 00:43:00,530 --> 00:43:02,400 what happens. 633 00:43:02,400 --> 00:43:05,300 So you see it's kind of like an abelian group, 634 00:43:05,300 --> 00:43:09,720 but there's an extra twist, so it's almost abelian, so 635 00:43:09,720 --> 00:43:13,990 the first step you can take away from abelian. 636 00:43:13,990 --> 00:43:16,220 And there's a way to quantify this notion. 637 00:43:16,220 --> 00:43:17,470 It's called "nilpotency." 638 00:43:17,470 --> 00:43:20,040 And we'll get to that in a second. 639 00:43:20,040 --> 00:43:25,070 But in particular, if you set S to be the following 640 00:43:25,070 --> 00:43:25,850 generators-- 641 00:43:32,490 --> 00:43:35,290 so if you take S to be these four elements, 642 00:43:35,290 --> 00:43:40,720 and you ask what does the r-th power of S look like, 643 00:43:40,720 --> 00:43:42,250 so I look at all the elements which 644 00:43:42,250 --> 00:43:48,010 can be written by r or at most r elements from S, 645 00:43:48,010 --> 00:43:49,960 what do these elements look like? 646 00:43:54,410 --> 00:43:55,350 What do you think? 647 00:43:55,350 --> 00:44:01,420 So if you look at elements in here, 648 00:44:01,420 --> 00:44:06,232 how large can this entry, the 1, comma, 2 entry be? 649 00:44:06,232 --> 00:44:07,080 r. 650 00:44:07,080 --> 00:44:10,140 So each time you do addition, so it's at most r. 651 00:44:10,140 --> 00:44:13,090 So let me be a bit rough here, and say it's big O of r. 652 00:44:13,090 --> 00:44:16,630 And likewise, the 2, 1, 2, 3, entry is also big O of r. 653 00:44:16,630 --> 00:44:19,660 What about the top right entry over here? 654 00:44:23,240 --> 00:44:25,380 So it grows like r squared, because there is 655 00:44:25,380 --> 00:44:27,170 an extra multiplication term. 656 00:44:29,790 --> 00:44:33,060 So you can be much more precise about the growth rate 657 00:44:33,060 --> 00:44:35,320 of these individual entries. 658 00:44:35,320 --> 00:44:39,150 But very roughly, it looks like this ball over here. 659 00:44:39,150 --> 00:44:47,230 So the size of S, the r-th ball of S, 660 00:44:47,230 --> 00:44:52,760 is roughly, it's on the order of 4th power of r. 661 00:44:55,650 --> 00:45:02,348 So in particular, the doubling constant, 662 00:45:02,348 --> 00:45:05,960 if r is reasonably large, is what? 663 00:45:10,440 --> 00:45:12,765 What happens when we go from r to 2r? 664 00:45:12,765 --> 00:45:16,185 The size increases by a factor of around 16. 665 00:45:21,610 --> 00:45:26,340 So that's an example of a set in a non-abelian group 666 00:45:26,340 --> 00:45:28,900 with bounded doubling, which is genuinely 667 00:45:28,900 --> 00:45:31,410 different from the examples we have seen so far. 668 00:45:31,410 --> 00:45:32,974 So that's non-abelian. 669 00:45:32,974 --> 00:45:33,474 Yeah. 670 00:45:33,474 --> 00:45:37,370 AUDIENCE: [INAUDIBLE] 671 00:45:37,370 --> 00:45:41,220 YUFEI ZHAO: The question is, is the size-- 672 00:45:41,220 --> 00:45:44,740 we've shown the size is-- 673 00:45:44,740 --> 00:45:47,280 I'm not being very precise here, but you can 674 00:45:47,280 --> 00:45:48,790 do upper bound and lower bound. 675 00:45:48,790 --> 00:45:51,810 So size turns out to be the order of r to the 4. 676 00:45:55,538 --> 00:45:57,330 So you want to show that there are actually 677 00:45:57,330 --> 00:45:59,290 enough elements over here that you can fill in, 678 00:45:59,290 --> 00:46:00,770 but I'll leave that to you. 679 00:46:08,218 --> 00:46:10,010 Can you build other examples like this one? 680 00:46:13,370 --> 00:46:14,810 Yeah. 681 00:46:14,810 --> 00:46:16,240 AUDIENCE: How do we know that this 682 00:46:16,240 --> 00:46:20,793 isn't similar to a co-set, the direct sum [INAUDIBLE]?? 683 00:46:20,793 --> 00:46:22,210 YUFEI ZHAO: Question is, how do we 684 00:46:22,210 --> 00:46:26,050 know this isn't like a co-set sum or a co-set progression? 685 00:46:26,050 --> 00:46:29,830 For one thing, this is not abelian. 686 00:46:29,830 --> 00:46:33,970 S, if you multiply entries of S in different orders, 687 00:46:33,970 --> 00:46:36,830 you get different elements. 688 00:46:36,830 --> 00:46:40,030 So already in that way, it's different from the examples 689 00:46:40,030 --> 00:46:41,135 that we have seen before. 690 00:46:41,135 --> 00:46:42,010 But no, you're right. 691 00:46:42,010 --> 00:46:44,740 So maybe we can write this a semi-direct product 692 00:46:44,740 --> 00:46:46,630 in terms of things we have seen before. 693 00:46:46,630 --> 00:46:49,600 And it is, in some sense, a semi-direct product, 694 00:46:49,600 --> 00:46:51,970 but it's a very special kind of semi-direct product. 695 00:46:58,202 --> 00:47:00,660 From that example, you can build bigger examples, of course 696 00:47:00,660 --> 00:47:03,060 with more entries in the matrix. 697 00:47:03,060 --> 00:47:06,480 But more generally, these things are 698 00:47:06,480 --> 00:47:09,270 what are known as "nilpotent groups." 699 00:47:09,270 --> 00:47:17,970 So that's an example of a nilpotent group. 700 00:47:17,970 --> 00:47:20,430 And to remind you, the definition of a nilpotent group 701 00:47:20,430 --> 00:47:23,700 is a group where the lower central series eventually 702 00:47:23,700 --> 00:47:24,360 terminates. 703 00:47:32,790 --> 00:47:36,500 In particular, inside that if you look at-- 704 00:47:36,500 --> 00:47:41,100 so this is the commutator of G, so look at all the elements 705 00:47:41,100 --> 00:47:44,208 that we recognize x, y, x inverse, 706 00:47:44,208 --> 00:47:46,750 y inverse-- the set of elements that can be written this way. 707 00:47:46,750 --> 00:47:48,240 So that's a subgroup. 708 00:47:48,240 --> 00:47:57,820 And if I repeat this operation enough times, 709 00:47:57,820 --> 00:48:02,340 I eventually would get just the identity. 710 00:48:02,340 --> 00:48:03,940 And you could trade on that group. 711 00:48:03,940 --> 00:48:09,150 If you do the commutator, so essentially you get rid 712 00:48:09,150 --> 00:48:15,630 of abelian-ness and you move up the whole diagonal, 713 00:48:15,630 --> 00:48:19,630 you create a commutator, you'd get rid of these-- 714 00:48:19,630 --> 00:48:20,650 all these two entries. 715 00:48:20,650 --> 00:48:23,790 So you get z alone. 716 00:48:23,790 --> 00:48:26,095 If you do it one more time, you zero out that entry. 717 00:48:33,710 --> 00:48:39,930 And so more generally, all of these nilpotent groups 718 00:48:39,930 --> 00:48:49,680 have this phenomenon, have the polynomial growth phenomenon. 719 00:48:55,180 --> 00:49:00,220 So if you take a set of generators and look at a ball, 720 00:49:00,220 --> 00:49:01,680 and look at the volume of the ball, 721 00:49:01,680 --> 00:49:04,210 how does the volume of the ball grow with the radius? 722 00:49:04,210 --> 00:49:07,160 It grows like a polynomial. 723 00:49:07,160 --> 00:49:09,250 And so let me define that. 724 00:49:09,250 --> 00:49:21,330 So given G, a finitely generated group, so generated by set S, 725 00:49:21,330 --> 00:49:36,560 we say that G has polynomial growth if the size S to the r 726 00:49:36,560 --> 00:49:40,350 grows like at most a polynomial in r. 727 00:49:48,470 --> 00:49:50,990 It's worth noting that this definition is really 728 00:49:50,990 --> 00:49:54,740 a definition about G. It does not depend 729 00:49:54,740 --> 00:49:56,487 on the choice of generators. 730 00:50:02,340 --> 00:50:04,790 You can have different choices, generators for the group. 731 00:50:04,790 --> 00:50:07,640 But if it has polynomial growth with respect 732 00:50:07,640 --> 00:50:10,210 to one set of generators, then it's the same. 733 00:50:10,210 --> 00:50:12,260 It also has polynomial growth with regards 734 00:50:12,260 --> 00:50:13,820 to every other set. 735 00:50:13,820 --> 00:50:18,970 So we've seen an example of groups with polynomial growth. 736 00:50:18,970 --> 00:50:21,118 Abelian groups have polynomial growth. 737 00:50:21,118 --> 00:50:22,660 So if you think of polynomial growth, 738 00:50:22,660 --> 00:50:25,690 think lattice or z to the m. 739 00:50:25,690 --> 00:50:27,930 So if you take a ball growing, so it 740 00:50:27,930 --> 00:50:34,480 has size growing like r to the dimension. 741 00:50:34,480 --> 00:50:36,600 But nilpotent groups is another example 742 00:50:36,600 --> 00:50:39,360 of groups with polynomial growth. 743 00:50:39,360 --> 00:50:42,600 And these are, intuitively at least for now, related 744 00:50:42,600 --> 00:50:44,707 to bounded doubling. 745 00:50:44,707 --> 00:50:47,040 If it's polynomial growth, then it has bounded doubling. 746 00:50:49,580 --> 00:50:52,840 So is there a classification of groups with bounded-- 747 00:50:52,840 --> 00:50:54,740 with polynomial growth? 748 00:50:54,740 --> 00:50:57,400 So if I tell you a group-- so an infinite group always, 749 00:50:57,400 --> 00:51:00,400 because otherwise if finite, then it maxes out already 750 00:51:00,400 --> 00:51:01,120 at some point. 751 00:51:01,120 --> 00:51:03,530 So I give you an infinite group. 752 00:51:03,530 --> 00:51:05,890 I tell you it has polynomial growth. 753 00:51:05,890 --> 00:51:07,648 What can you tell me about this group? 754 00:51:07,648 --> 00:51:09,190 Is there some characterization that's 755 00:51:09,190 --> 00:51:12,133 an inverse of what we've seen so far? 756 00:51:12,133 --> 00:51:13,050 And the answer is yes. 757 00:51:13,050 --> 00:51:16,060 And this is a famous and deep result of Gromov. 758 00:51:19,290 --> 00:51:25,180 So Gromov's theorem on groups of polynomial growth 759 00:51:25,180 --> 00:51:26,660 from the '80s. 760 00:51:26,660 --> 00:51:33,910 Gromov showed that a finitely generated 761 00:51:33,910 --> 00:51:42,400 group has polynomial growth if and only 762 00:51:42,400 --> 00:51:55,230 if it's virtually nilpotent, where "virtually" 763 00:51:55,230 --> 00:52:01,200 is an adverb in group theory where 764 00:52:01,200 --> 00:52:04,950 you have some property like "abelian," or "solvable," 765 00:52:04,950 --> 00:52:06,030 or whatever. 766 00:52:06,030 --> 00:52:16,490 So virtually P means that there exists a finite index subgroup 767 00:52:16,490 --> 00:52:24,950 with property P. So "virtually nilpotent" 768 00:52:24,950 --> 00:52:29,100 means there is a finite index subgroup that is nilpotent. 769 00:52:29,100 --> 00:52:34,140 So it completely characterizes groups of polynomial growth. 770 00:52:34,140 --> 00:52:37,080 So basically, all the examples we've seen so far 771 00:52:37,080 --> 00:52:43,680 are representative, so up to changing by a finite index 772 00:52:43,680 --> 00:52:45,780 subgroup, which as you would expect, 773 00:52:45,780 --> 00:52:50,520 shouldn't change the growth nature by so much. 774 00:52:50,520 --> 00:52:53,730 There are some analogies to be made here with, 775 00:52:53,730 --> 00:52:59,130 for example in geometry, you ask in Euclidean space, 776 00:52:59,130 --> 00:53:03,560 how fast is the ball of radius r growing? 777 00:53:03,560 --> 00:53:07,580 In dimension d, it grows like r to the d. 778 00:53:07,580 --> 00:53:10,920 What about in the hyperbolic space? 779 00:53:10,920 --> 00:53:13,530 Does anyone know how fast, in a hyperbolic space, 780 00:53:13,530 --> 00:53:15,570 a ball of radius r grows? 781 00:53:19,130 --> 00:53:23,100 It's exponential in the radius. 782 00:53:23,100 --> 00:53:27,390 So for non-negatively curved spaces, 783 00:53:27,390 --> 00:53:30,240 the balls grow polynomially. 784 00:53:30,240 --> 00:53:33,720 But for something that's negatively curvatured, 785 00:53:33,720 --> 00:53:35,970 in particular the hyperbolic space, 786 00:53:35,970 --> 00:53:39,940 the ball growth might be exponential. 787 00:53:39,940 --> 00:53:43,190 You have a similar phenomenon happening here. 788 00:53:43,190 --> 00:53:44,870 The opposite of polynomial growth 789 00:53:44,870 --> 00:53:46,400 is, well, super polynomial growth, 790 00:53:46,400 --> 00:53:53,050 but one specific example is that of a free group, where 791 00:53:53,050 --> 00:53:56,730 there are no relations between the generators. 792 00:53:56,730 --> 00:54:03,930 In that case, the balls, they grow like exponentially. 793 00:54:03,930 --> 00:54:08,550 So the balls grow exponentially in the radius. 794 00:54:08,550 --> 00:54:11,470 Gromov's theorem is a deep theorem. 795 00:54:11,470 --> 00:54:16,150 And its original proof used some very hard tools 796 00:54:16,150 --> 00:54:17,830 coming from geometry. 797 00:54:17,830 --> 00:54:23,230 And Gromov developed a notion of convergence of metric spaces, 798 00:54:23,230 --> 00:54:26,740 somewhat akin to our discussion of graph limits. 799 00:54:26,740 --> 00:54:28,960 So starting with discrete objects, 800 00:54:28,960 --> 00:54:32,890 he looked at some convergence to some continuous objects, 801 00:54:32,890 --> 00:54:37,420 and then used some very deep results 802 00:54:37,420 --> 00:54:41,740 from the classification of locally compact groups 803 00:54:41,740 --> 00:54:44,410 to derive this result over here. 804 00:54:48,060 --> 00:54:50,840 So this proof has been quite influential, 805 00:54:50,840 --> 00:54:53,930 and is related to something called 806 00:54:53,930 --> 00:55:05,640 "Hilbert's fifth problem, which concerns characterizations 807 00:55:05,640 --> 00:55:07,336 of Lie groups. 808 00:55:07,336 --> 00:55:09,620 So all of these are inverse-type problems. 809 00:55:09,620 --> 00:55:11,690 I tell you some structure has some property. 810 00:55:11,690 --> 00:55:15,770 Describe that structure. 811 00:55:15,770 --> 00:55:20,030 What does this all have to do with Freiman's theorem? 812 00:55:20,030 --> 00:55:21,310 Already you see some relation. 813 00:55:21,310 --> 00:55:23,602 So there seems, at least intuitively, some relationship 814 00:55:23,602 --> 00:55:26,270 between groups of polynomial growth versus subsets 815 00:55:26,270 --> 00:55:27,650 of bounded doubling. 816 00:55:27,650 --> 00:55:31,320 One implies the other, although not in the converse. 817 00:55:31,320 --> 00:55:32,570 And they are indeed related. 818 00:55:32,570 --> 00:55:35,720 And this comes out of some very recent work. 819 00:55:35,720 --> 00:55:37,790 I should also mention that Gromov's theorem has 820 00:55:37,790 --> 00:55:40,730 been made simplified by Kleiner, who 821 00:55:40,730 --> 00:55:44,930 gave an important simplification, a more 822 00:55:44,930 --> 00:55:47,080 elementary proof of Gromov's theorem. 823 00:55:49,790 --> 00:55:52,160 So let's talk about the non-abelian version 824 00:55:52,160 --> 00:55:53,330 of Freiman's theorem. 825 00:55:59,630 --> 00:56:02,270 We would like some result that says 826 00:56:02,270 --> 00:56:10,220 that is it true that every set, most every set of-- 827 00:56:10,220 --> 00:56:12,705 so previously, we had small doubling. 828 00:56:12,705 --> 00:56:15,080 You want to have some similar notion, although it may not 829 00:56:15,080 --> 00:56:19,330 be exactly small doubling, but let me not be very precise 830 00:56:19,330 --> 00:56:24,043 and to say, "small doubling." 831 00:56:24,043 --> 00:56:26,210 In literature, these things are sometimes also known 832 00:56:26,210 --> 00:56:27,770 as "approximate groups." 833 00:56:32,027 --> 00:56:34,930 So if you look this up, you will get to the relevant literature 834 00:56:34,930 --> 00:56:36,250 on the subject. 835 00:56:36,250 --> 00:56:39,160 Most every set of small doubling in some non-abelian group 836 00:56:39,160 --> 00:56:45,670 behaves like one of these known examples, something 837 00:56:45,670 --> 00:56:58,160 which is some combination of subgroups and nilpotent balls. 838 00:57:03,300 --> 00:57:06,240 So these combinations are sometimes known as "co-set 839 00:57:06,240 --> 00:57:07,729 nilprogressions." 840 00:57:15,560 --> 00:57:18,910 So this was something that was only 841 00:57:18,910 --> 00:57:21,190 explored in the past 10 years or so 842 00:57:21,190 --> 00:57:25,540 in a series of very difficult works. 843 00:57:25,540 --> 00:57:27,940 Previously, it had been known, and still 844 00:57:27,940 --> 00:57:30,190 was being investigated for various special classes 845 00:57:30,190 --> 00:57:32,800 of matrix groups or special classes of groups 846 00:57:32,800 --> 00:57:35,850 like solvable groups and whatnot, 847 00:57:35,850 --> 00:57:38,900 that are more explicit or easier to handle 848 00:57:38,900 --> 00:57:40,730 or closer to the abelian analog. 849 00:57:43,580 --> 00:57:50,370 There was important work of Hrushovski, 850 00:57:50,370 --> 00:57:55,090 which was published about 10 years ago, 851 00:57:55,090 --> 00:57:58,320 who showed using model theory techniques, 852 00:57:58,320 --> 00:58:05,700 so using methods from logic, that a weak version 853 00:58:05,700 --> 00:58:12,970 of Freiman's theorem is true for non-abelian groups. 854 00:58:12,970 --> 00:58:22,842 And later on, Breuillard, Green, and Tao building on Hrushovskis 855 00:58:22,842 --> 00:58:24,800 work-- so this actually came quite a bit later, 856 00:58:24,800 --> 00:58:26,730 even though the journal publication dates are the same 857 00:58:26,730 --> 00:58:27,380 year-- 858 00:58:27,380 --> 00:58:31,430 so they were able to build on Hrushovski's work, 859 00:58:31,430 --> 00:58:35,480 and greatly expanding on it, and going back to some of the older 860 00:58:35,480 --> 00:58:40,070 techniques coming from Hilbert's fifth problem, and as a result, 861 00:58:40,070 --> 00:58:43,580 proved an inverse structure theorem 862 00:58:43,580 --> 00:58:46,760 that gave some kind of answer to this question 863 00:58:46,760 --> 00:58:48,830 of non-abelian Freiman. 864 00:58:48,830 --> 00:58:51,380 So we now do have some theorem which 865 00:58:51,380 --> 00:58:54,320 is like Freiman's theorem for abelian groups that 866 00:58:54,320 --> 00:58:57,560 says in a non-abelian group, if you have something that 867 00:58:57,560 --> 00:59:02,060 resembles small doubling, then the set must, in some sense, 868 00:59:02,060 --> 00:59:06,680 look like a combination of subgroups and nilpotent balls. 869 00:59:06,680 --> 00:59:09,350 But let me not be precise at all. 870 00:59:09,350 --> 00:59:10,940 The methods here build on Hrushovski. 871 00:59:10,940 --> 00:59:14,930 And Hrushovski used model theory, which is kind of-- 872 00:59:14,930 --> 00:59:19,640 it's something where-- in particular, 873 00:59:19,640 --> 00:59:21,560 one feature of all of these proofs 874 00:59:21,560 --> 00:59:22,910 is that they give no bounds. 875 00:59:26,950 --> 00:59:29,650 Similar to what we've seen earlier in the course, 876 00:59:29,650 --> 00:59:33,790 in proofs that involved compactness, what happens 877 00:59:33,790 --> 00:59:36,055 here is that the arguments use ultra filters. 878 00:59:39,420 --> 00:59:43,190 So there are these constructions from mathematical logic. 879 00:59:43,190 --> 00:59:46,340 And like compactness, they give no bounds. 880 00:59:46,340 --> 00:59:49,720 So it remains an open problem to prove Freiman's theorem 881 00:59:49,720 --> 00:59:52,420 for non-abelian groups with some concrete bounds. 882 00:59:52,420 --> 00:59:53,197 Question. 883 00:59:53,197 --> 00:59:57,230 AUDIENCE: [INAUDIBLE] nilpotent ball? 884 00:59:57,230 --> 00:59:59,480 YUFEI ZHAO: What is nilpotent ball? 885 00:59:59,480 --> 01:00:01,610 I don't want to give a precise definition, 886 01:00:01,610 --> 01:00:04,610 but roughly speaking, it's balls that 887 01:00:04,610 --> 01:00:07,790 come out of those types of constructions. 888 01:00:07,790 --> 01:00:09,610 So you take a nilpotent subgroup. 889 01:00:09,610 --> 01:00:11,030 You take a nilpotent group. 890 01:00:11,030 --> 01:00:13,400 You look at an image of a nilpotent group 891 01:00:13,400 --> 01:00:23,640 into your group, and then look at the image of that ball, so 892 01:00:23,640 --> 01:00:29,130 something that looks like one of the previous constructions. 893 01:00:29,130 --> 01:00:31,495 So that's all I want to say about non-abelian extensions 894 01:00:31,495 --> 01:00:32,370 of Freiman's theorem. 895 01:00:32,370 --> 01:00:33,598 Any questions? 896 01:00:33,598 --> 01:00:35,140 AUDIENCE: Would you say one more time 897 01:00:35,140 --> 01:00:37,408 what you mean by "approximate group?" 898 01:00:37,408 --> 01:00:38,700 YUFEI ZHAO: So what I mean by-- 899 01:00:38,700 --> 01:00:41,670 you can look in the papers and see the precise definitions, 900 01:00:41,670 --> 01:00:47,700 but roughly speaking, it's that if you have-- 901 01:00:47,700 --> 01:00:51,090 there are different kinds of definitions and most of them 902 01:00:51,090 --> 01:00:51,930 are equivalent. 903 01:00:51,930 --> 01:00:54,120 But one version is that you have a set A 904 01:00:54,120 --> 01:01:05,940 such that A is coverable by K translates of A, 905 01:01:05,940 --> 01:01:08,760 so it's a bit more than just the size information, 906 01:01:08,760 --> 01:01:11,770 but it's actually related to size information. 907 01:01:11,770 --> 01:01:16,060 So we've already seen in this course how many of these 908 01:01:16,060 --> 01:01:18,060 different notions can go back and forth from one 909 01:01:18,060 --> 01:01:21,040 to the other, covering to size, and whatnot. 910 01:01:27,110 --> 01:01:28,880 The final thing I want to discuss today 911 01:01:28,880 --> 01:01:31,780 is one of the most central open problems 912 01:01:31,780 --> 01:01:35,230 in additive combinatorics going back to the abelian version. 913 01:01:35,230 --> 01:01:37,930 So this is known as the "polynomial Freiman-Ruzsa 914 01:01:37,930 --> 01:01:38,933 conjecture." 915 01:01:48,600 --> 01:01:52,050 So we would like some kind of a Freiman theorem 916 01:01:52,050 --> 01:01:56,370 that preserves the constants up to polynomial changes 917 01:01:56,370 --> 01:01:59,788 without losing an exponent. 918 01:01:59,788 --> 01:02:02,980 Now, from earlier discussions, I showed you 919 01:02:02,980 --> 01:02:09,950 that the bounds that we almost proved is close to the truth. 920 01:02:09,950 --> 01:02:11,710 You do need some kind of exponential loss 921 01:02:11,710 --> 01:02:14,975 in the blowup size of the GAP. 922 01:02:14,975 --> 01:02:16,600 But it turns out those kind of examples 923 01:02:16,600 --> 01:02:18,100 are slightly misleading. 924 01:02:18,100 --> 01:02:20,800 So let's look at the examples of the constructions again. 925 01:02:20,800 --> 01:02:26,500 So if A-- so just for simplicity in exposition, 926 01:02:26,500 --> 01:02:31,550 I'm going to stick with F2 to the n, at least initially. 927 01:02:31,550 --> 01:02:41,090 So if A is an independent set of size n, 928 01:02:41,090 --> 01:02:45,660 then K, being the doubling constant of A, 929 01:02:45,660 --> 01:02:48,650 is roughly like n over 2. 930 01:02:48,650 --> 01:02:52,145 And yet the subgroup that contains A 931 01:02:52,145 --> 01:03:00,110 has size 2 to the something on the order of K times A. 932 01:03:00,110 --> 01:03:04,830 So you necessarily incur an exponential loss over here. 933 01:03:04,830 --> 01:03:08,740 Now, you might complain that the size of A here is basically K. 934 01:03:08,740 --> 01:03:11,570 But of course, I can blow up this example 935 01:03:11,570 --> 01:03:17,600 by considering what happens if you take each element here, 936 01:03:17,600 --> 01:03:19,610 and blow it up into an entire subspace. 937 01:03:25,110 --> 01:03:28,150 So the e's are the coordinate vectors. 938 01:03:28,150 --> 01:03:32,340 So now I'm sitting inside F2 to the m plus n. 939 01:03:32,340 --> 01:03:33,690 And that gives me this set. 940 01:03:36,800 --> 01:03:39,220 The doubling constant is still the same as before. 941 01:03:44,900 --> 01:03:54,270 And yet, we see that the subgroup generated by A 942 01:03:54,270 --> 01:03:56,310 still has this exponential blowup 943 01:03:56,310 --> 01:04:00,630 in this constant, exponential in the doubling constant. 944 01:04:00,630 --> 01:04:04,050 But now you see in this example here, 945 01:04:04,050 --> 01:04:07,320 even though the subgroup generated by A 946 01:04:07,320 --> 01:04:11,190 can be much larger than A, so everything's still constant, so 947 01:04:11,190 --> 01:04:14,100 much larger in terms of as a function of the doubling 948 01:04:14,100 --> 01:04:18,740 constant, A has a very large structure. 949 01:04:18,740 --> 01:04:25,070 So A contains a very large subspace. 950 01:04:30,625 --> 01:04:32,685 By "subspace," I mean affine subspace. 951 01:04:36,300 --> 01:04:40,920 And the subspace here is comparable to the size 952 01:04:40,920 --> 01:04:41,850 of A itself. 953 01:04:44,900 --> 01:04:47,980 So you might wonder, if you don't 954 01:04:47,980 --> 01:04:52,540 care about containing A inside a single subspace, 955 01:04:52,540 --> 01:04:56,920 can you do much better in terms of bounds? 956 01:04:56,920 --> 01:04:59,170 And that's the content of the polynomial Freiman-Ruzsa 957 01:04:59,170 --> 01:05:01,030 conjecture. 958 01:05:01,030 --> 01:05:07,030 The PFR conjecture for F2 to the m 959 01:05:07,030 --> 01:05:13,609 says that if you have a subset of F2 to the m 960 01:05:13,609 --> 01:05:21,010 and A plus A is size at most K times the size of A, 961 01:05:21,010 --> 01:05:28,840 then there exists a subspace V of size 962 01:05:28,840 --> 01:05:44,880 at most A such that V contains a large proportion of A. 963 01:05:44,880 --> 01:05:46,350 And the large here-- 964 01:05:46,350 --> 01:05:50,220 we only lose something that is polynomial in these doubling 965 01:05:50,220 --> 01:05:51,162 constants. 966 01:05:54,120 --> 01:05:55,540 So that's the case. 967 01:05:55,540 --> 01:05:57,300 It's over here. 968 01:05:57,300 --> 01:06:04,160 So instead of containing A inside an entire subspace, 969 01:06:04,160 --> 01:06:07,850 I just want to contain a large fraction of A in a subspace. 970 01:06:07,850 --> 01:06:09,740 And the conjecture is that I do not 971 01:06:09,740 --> 01:06:12,725 need to incur exponential losses in the constants. 972 01:06:12,725 --> 01:06:15,520 AUDIENCE: So V is an affine subspace? 973 01:06:15,520 --> 01:06:18,390 YUFEI ZHAO: V is-- 974 01:06:18,390 --> 01:06:20,552 question is, V is an affine subspace. 975 01:06:20,552 --> 01:06:22,260 You can think of V as an affine subspace. 976 01:06:22,260 --> 01:06:23,910 You can think of V as a subspace. 977 01:06:23,910 --> 01:06:26,179 It doesn't actually matter in this formulation. 978 01:06:35,170 --> 01:06:36,860 There's an equivalent formulation 979 01:06:36,860 --> 01:06:41,660 which you might like better, where you might complain, 980 01:06:41,660 --> 01:06:43,382 initially, PFR is initially-- 981 01:06:43,382 --> 01:06:44,840 Freiman's theorem is about covering 982 01:06:44,840 --> 01:06:49,005 A. And now we've only covered a part of A. 983 01:06:49,005 --> 01:06:50,880 But of course, we saw from earlier arguments, 984 01:06:50,880 --> 01:06:53,160 you can use Ruzsa's covering lemma to go 985 01:06:53,160 --> 01:06:56,890 from covering a part of A to covering all of A. 986 01:06:56,890 --> 01:07:00,370 Indeed, it this the case that this formulation 987 01:07:00,370 --> 01:07:04,240 is equivalent to the formulation that if A 988 01:07:04,240 --> 01:07:14,300 is in F to the n and A plus A size at most K times A, 989 01:07:14,300 --> 01:07:23,150 then there exists some subspace V with the size of V no larger 990 01:07:23,150 --> 01:07:28,010 than the size of A, such that A can 991 01:07:28,010 --> 01:07:43,100 be covered by polynomial in K many co-sets of V. 992 01:07:43,100 --> 01:07:46,130 We see that here. 993 01:07:46,130 --> 01:07:52,790 Here A has doubling constant K, which is around the same as n. 994 01:07:52,790 --> 01:07:58,760 And even though I cannot contain A by a single subspace 995 01:07:58,760 --> 01:08:04,710 of roughly the same size, I can use K different translates 996 01:08:04,710 --> 01:08:14,440 to cover A. Any questions? 997 01:08:19,340 --> 01:08:22,460 So I want to leave it to you as an exercise 998 01:08:22,460 --> 01:08:27,790 to prove that these two versions are equivalent to each other. 999 01:08:27,790 --> 01:08:29,154 It's not too hard. 1000 01:08:29,154 --> 01:08:31,510 It's something if I had more time, I would show you. 1001 01:08:31,510 --> 01:08:37,279 It uses Ruzsa covering lemma to prove this equivalence. 1002 01:08:39,859 --> 01:08:41,220 The nice thing about the-- 1003 01:08:41,220 --> 01:08:44,180 so the polynomial Freiman-Ruzsa conjecture, PFR conjecture, 1004 01:08:44,180 --> 01:08:46,609 is considered a central conjecture 1005 01:08:46,609 --> 01:08:51,140 in additive combinatorics, because it has many equivalent 1006 01:08:51,140 --> 01:08:54,590 formulations and relates to many problems that 1007 01:08:54,590 --> 01:08:56,990 are central to the subject. 1008 01:08:56,990 --> 01:08:59,720 So we would like some kind of an inverse theorem that gives you 1009 01:08:59,720 --> 01:09:01,220 these polynomial bounds. 1010 01:09:01,220 --> 01:09:06,640 And I'll mention a couple of these equivalent formulations. 1011 01:09:06,640 --> 01:09:10,200 Here is an equivalent formulation 1012 01:09:10,200 --> 01:09:16,700 which is rather attractive, where instead of considering 1013 01:09:16,700 --> 01:09:18,859 subsets, we're going to formulate 1014 01:09:18,859 --> 01:09:22,354 something that has to do with approximate homomorphisms. 1015 01:09:27,310 --> 01:09:32,010 So the statement still conjecture 1016 01:09:32,010 --> 01:09:37,950 is that if F is a function from a Boolean space 1017 01:09:37,950 --> 01:09:45,380 to another Boolean space is such that F is approximately 1018 01:09:45,380 --> 01:09:52,727 a homomorphism in the sense that the set of possible errors-- 1019 01:09:52,727 --> 01:09:54,480 so if it's actually a homomorphism, 1020 01:09:54,480 --> 01:09:56,460 then this quantity is always equal to 0-- 1021 01:09:56,460 --> 01:09:59,760 but it's approximately a homomorphism in the sense 1022 01:09:59,760 --> 01:10:11,140 that the set of such errors is bounded by K in size, 1023 01:10:11,140 --> 01:10:14,730 the conclusion, the conjecture claims that then there 1024 01:10:14,730 --> 01:10:24,650 exists an actual homomorphism, an actual linear map G, 1025 01:10:24,650 --> 01:10:32,810 such that F is very close to G, as 1026 01:10:32,810 --> 01:10:40,450 in that the set of possible discrepancies between F and G 1027 01:10:40,450 --> 01:10:52,890 is bounded, where you only lose at most a polynomial in K. 1028 01:10:52,890 --> 01:10:55,600 So if you are an approximate homomorphism in this sense, 1029 01:10:55,600 --> 01:11:00,500 then you are actually very close to an actual linear map. 1030 01:11:00,500 --> 01:11:06,050 Now, it is not too hard to prove a much quantitatively weaker 1031 01:11:06,050 --> 01:11:07,580 version of this statement. 1032 01:11:07,580 --> 01:11:15,985 So I claim that it is trivial to show upper bound of at most 2 1033 01:11:15,985 --> 01:11:20,860 to the K over here. 1034 01:11:20,860 --> 01:11:22,300 So think about that. 1035 01:11:22,300 --> 01:11:25,390 So if I give you an F, I can just 1036 01:11:25,390 --> 01:11:30,050 think about what the values of F are on the basis, 1037 01:11:30,050 --> 01:11:32,660 and extend it to a linear map. 1038 01:11:35,570 --> 01:11:44,160 Then this set is necessarily a span of that set, 1039 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 1040 01:11:51,150 --> 01:12:01,870 have to lose a polynomial in K. 1041 01:12:01,870 --> 01:12:06,890 There is also a version of the polynomial Freiman-Ruzsa 1042 01:12:06,890 --> 01:12:12,500 conjecture which is related to things we've discussed earlier 1043 01:12:12,500 --> 01:12:15,970 regarding Szemeredi's theorem. 1044 01:12:15,970 --> 01:12:20,680 And in fact, the polynomial Freiman-Ruzsa conjecture kind 1045 01:12:20,680 --> 01:12:24,820 of came back into popularity partly because 1046 01:12:24,820 --> 01:12:28,400 of Gowers' proof of Szemeredi's theorem 1047 01:12:28,400 --> 01:12:31,260 that used many of these tools. 1048 01:12:31,260 --> 01:12:34,250 So let me state it here. 1049 01:12:34,250 --> 01:12:41,320 So we've seen some statement like this in an earlier 1050 01:12:41,320 --> 01:12:48,670 lecture, but not very precisely or not precisely in this form. 1051 01:12:48,670 --> 01:12:53,600 And I won't define for you all the notation here, 1052 01:12:53,600 --> 01:12:57,050 but hopefully, you get a rough sense of what it's about. 1053 01:12:57,050 --> 01:12:59,200 So we want some kind of an inverse statement 1054 01:12:59,200 --> 01:13:03,670 for what's known as a "quadratic uniformity norm," 1055 01:13:03,670 --> 01:13:05,380 "quadratic Gowers' uniformity norm." 1056 01:13:12,700 --> 01:13:17,020 So recall back to our discussion of the proof of Roth's theorem, 1057 01:13:17,020 --> 01:13:20,050 the Fourier analytic proof of Roth's theorem. 1058 01:13:20,050 --> 01:13:21,820 We want to say that-- 1059 01:13:21,820 --> 01:13:26,590 but now think about not three APs, but four APs. 1060 01:13:26,590 --> 01:13:32,910 So we want to know if you have a function F on the Boolean cube, 1061 01:13:32,910 --> 01:13:43,667 and this function is 1 bounded, and-- 1062 01:13:43,667 --> 01:13:45,500 I'm going to write down some notation, which 1063 01:13:45,500 --> 01:13:47,190 we are not going to define-- 1064 01:13:47,190 --> 01:13:54,880 but the Gowers' u3 norm is at least some delta. 1065 01:13:54,880 --> 01:13:58,570 So this is something which is related to 4 AP counts. 1066 01:13:58,570 --> 01:14:00,868 So in particular, if this number is small, 1067 01:14:00,868 --> 01:14:03,160 then you have a counting lemma for four-term arithmetic 1068 01:14:03,160 --> 01:14:06,510 progressions. 1069 01:14:06,510 --> 01:14:17,330 If this is true, then there exists a quadratic polynomial q 1070 01:14:17,330 --> 01:14:27,520 in n variables over F2 such that your function 1071 01:14:27,520 --> 01:14:34,660 F correlates with this quadratic exponential in q. 1072 01:14:34,660 --> 01:14:44,750 And the correlation here is something where you only lose 1073 01:14:44,750 --> 01:14:48,320 a polynomial in the parameters. 1074 01:14:48,320 --> 01:14:49,830 So previously, I quoted something 1075 01:14:49,830 --> 01:14:55,410 where you lose something that's only a constant in delta, 1076 01:14:55,410 --> 01:14:56,400 and that is true. 1077 01:14:56,400 --> 01:14:57,410 That is known. 1078 01:14:57,410 --> 01:15:00,940 But we believe, so it's conjecture, 1079 01:15:00,940 --> 01:15:03,650 that you only lose a polynomial in these parameters. 1080 01:15:03,650 --> 01:15:05,300 So this type of statement-- remember, 1081 01:15:05,300 --> 01:15:07,800 in our proof of Roth's theorem, something like this came up. 1082 01:15:07,800 --> 01:15:10,120 So something like this came up as a crucial step 1083 01:15:10,120 --> 01:15:11,440 in the proof of Roth's theorem. 1084 01:15:11,440 --> 01:15:17,680 If you have something where you look at counting lemma, 1085 01:15:17,680 --> 01:15:19,180 and you exhibit something like this, 1086 01:15:19,180 --> 01:15:22,110 then you can exhibit a large Fourier character. 1087 01:15:22,110 --> 01:15:24,130 And in higher order Fourier analysis, 1088 01:15:24,130 --> 01:15:27,850 something like this corresponds to having 1089 01:15:27,850 --> 01:15:29,541 a large Fourier transform. 1090 01:15:32,310 --> 01:15:34,440 It turns out that all of these formulations 1091 01:15:34,440 --> 01:15:36,540 of polynomial Freiman-Ruzsa conjecture 1092 01:15:36,540 --> 01:15:40,410 are equivalent to each other. 1093 01:15:40,410 --> 01:15:45,420 And they're all equivalent in a very quantitative sense, 1094 01:15:45,420 --> 01:15:53,840 so up to polynomial changes in the bounds. 1095 01:15:57,563 --> 01:15:58,980 So in particular, if you can prove 1096 01:15:58,980 --> 01:16:01,230 some bound for some version, then that automatically 1097 01:16:01,230 --> 01:16:04,320 leads to bounds for the other versions. 1098 01:16:04,320 --> 01:16:06,870 The proof of equivalences is not trivial, 1099 01:16:06,870 --> 01:16:12,210 but it's also not too complicated. 1100 01:16:12,210 --> 01:16:15,630 It takes some work, but it's not too complicated. 1101 01:16:15,630 --> 01:16:19,680 The best bounds for the polynomial Freiman-Ruzsa 1102 01:16:19,680 --> 01:16:22,500 conjecture, and hence for all of these versions, 1103 01:16:22,500 --> 01:16:26,100 is again due to Tom Sanders. 1104 01:16:26,100 --> 01:16:45,360 And he proved a version of PFR with quasi-polynomial bounds, 1105 01:16:45,360 --> 01:16:48,310 where by "quasi-polynomial bounds," I mean, 1106 01:16:48,310 --> 01:16:53,430 for instance over here, instead of K. 1107 01:16:53,430 --> 01:17:02,650 He proved it for something which is like e to the poly log K, 1108 01:17:02,650 --> 01:17:07,980 so like K to the log K, but K to the poly log K. 1109 01:17:07,980 --> 01:17:11,410 So it's almost polynomial, but not quite there. 1110 01:17:14,180 --> 01:17:17,570 And it's considered a central open problem 1111 01:17:17,570 --> 01:17:22,010 to better understand the polynomial Freiman-Ruzsa 1112 01:17:22,010 --> 01:17:24,150 conjecture. 1113 01:17:24,150 --> 01:17:26,010 And we believe that this is something 1114 01:17:26,010 --> 01:17:30,330 that could lead to a lot of important new tools 1115 01:17:30,330 --> 01:17:32,340 and techniques that are relevant to the rest 1116 01:17:32,340 --> 01:17:35,016 of additive combinatorics. 1117 01:17:35,016 --> 01:17:37,330 Yeah. 1118 01:17:37,330 --> 01:17:39,790 AUDIENCE: Using the fact that all of these are equivalent, 1119 01:17:39,790 --> 01:17:43,000 is it possible to get a proof of Freiman's theorem using 1120 01:17:43,000 --> 01:17:46,062 the bound of 2 to the K to be approximate [INAUDIBLE]?? 1121 01:17:46,062 --> 01:17:47,520 YUFEI ZHAO: OK, so the question is, 1122 01:17:47,520 --> 01:17:51,890 we know that that up there has 2 to the K, 1123 01:17:51,890 --> 01:17:56,550 so you're asking can you use this 2 to the K 1124 01:17:56,550 --> 01:18:00,790 to get some bound for polynomial, 1125 01:18:00,790 --> 01:18:03,130 for something like this? 1126 01:18:03,130 --> 01:18:04,380 And the answer is yes. 1127 01:18:04,380 --> 01:18:07,250 So you can use that proof to go through some proofs 1128 01:18:07,250 --> 01:18:09,030 and get here. 1129 01:18:09,030 --> 01:18:12,870 I don't remember how this equivalence goes, but remember 1130 01:18:12,870 --> 01:18:16,920 that the proof of Freiman's theorem for F2 to the n 1131 01:18:16,920 --> 01:18:19,260 wasn't so hard. 1132 01:18:19,260 --> 01:18:22,500 So we didn't use very many tools. 1133 01:18:22,500 --> 01:18:24,870 Unfortunately, I don't have time to tell you 1134 01:18:24,870 --> 01:18:29,070 the formulations of polynomial Freiman-Ruzsa conjecture 1135 01:18:29,070 --> 01:18:33,330 over the integers, and also over arbitrary abelian groups. 1136 01:18:33,330 --> 01:18:36,480 But there are formulations over the integers, 1137 01:18:36,480 --> 01:18:40,140 and that's one that people care just as much about. 1138 01:18:40,140 --> 01:18:42,990 And there are also different equivalent versions, 1139 01:18:42,990 --> 01:18:47,457 but things are a bit nicer in the Boolean case. 1140 01:18:47,457 --> 01:18:48,894 Yeah. 1141 01:18:48,894 --> 01:18:50,331 AUDIENCE: You said [INAUDIBLE]? 1142 01:18:52,496 --> 01:18:54,621 YUFEI ZHAO: I'm sorry, can you repeat the question? 1143 01:18:54,621 --> 01:18:58,389 AUDIENCE: [INAUDIBLE]. 1144 01:18:58,389 --> 01:18:59,472 Yeah, what does that mean? 1145 01:18:59,472 --> 01:19:01,431 YUFEI ZHAO: Are you asking what does this mean? 1146 01:19:01,431 --> 01:19:02,163 AUDIENCE: Yeah. 1147 01:19:02,163 --> 01:19:04,580 YUFEI ZHAO: So this is what's called a "Gowers' uniformity 1148 01:19:04,580 --> 01:19:06,090 norm." 1149 01:19:06,090 --> 01:19:13,820 So something I encourage you to look up. 1150 01:19:13,820 --> 01:19:17,790 In fact, there is an unassigned problem in the problem set 1151 01:19:17,790 --> 01:19:21,150 that's related to the Gowers' uniformity norm 1152 01:19:21,150 --> 01:19:24,610 before you U2, which just relates to Fourier analysis. 1153 01:19:24,610 --> 01:19:29,500 But U3 is related to 4 AP's and quadratic Fourier analysis.