[{"text": "[SQUEAKING]", "start": 0.0, "duration": 1.952}, {"text": "[PAPER RUSTLING]", "start": 1.952, "duration": 1.452}, {"text": "[CLICKING]", "start": 3.904, "duration": 0.976}, {"text": "YUFEI ZHAO: Last time we started\ntalking about Roth's theorem,", "start": 18.08, "duration": 2.88}, {"text": "and we showed a Fourier\nanalytic proof of Roth's theorem", "start": 20.96, "duration": 5.46}, {"text": "in the finite field model.", "start": 26.42, "duration": 2.34}, {"text": "So Roth's theorem\nin F3 to the N.", "start": 28.76, "duration": 4.01}, {"text": "And I want to today show\nyou how to modify that proof", "start": 32.77, "duration": 4.9}, {"text": "to work in integers.", "start": 37.67, "duration": 1.98}, {"text": "And this will be basically\nRoth's original proof", "start": 39.65, "duration": 6.48}, {"text": "of his theorem.", "start": 46.13, "duration": 0.69}, {"text": "OK.", "start": 56.44, "duration": 0.97}, {"text": "So what we'll prove\ntoday is the statement", "start": 57.41, "duration": 3.45}, {"text": "that the size of the largest\n3AP-free subset of 1 through N", "start": 60.86, "duration": 9.72}, {"text": "is, at most, N divided\nby log log N. OK,", "start": 70.58, "duration": 10.5}, {"text": "so we'll prove a\nbound of this form.", "start": 81.08, "duration": 1.8}, {"text": "The strategy of this proof\nwill be very similar to the one", "start": 86.45, "duration": 3.15}, {"text": "that we had from last time.", "start": 89.6, "duration": 1.32}, {"text": "So let me review for you\nwhat is the strategy.", "start": 90.92, "duration": 1.917}, {"text": "So from last time, the\nproof had three main steps.", "start": 104.68, "duration": 4.85}, {"text": "In the first step, we\nobserved that if you", "start": 109.53, "duration": 3.39}, {"text": "are in the 3AP-free set then\nthere exists a large Fourier", "start": 112.92, "duration": 9.25}, {"text": "coefficient.", "start": 122.17, "duration": 1.104}, {"text": "From this Fourier\ncoefficient, we", "start": 131.81, "duration": 1.92}, {"text": "were able to extract\na large subspace where", "start": 133.73, "duration": 8.64}, {"text": "there is a density increment.", "start": 142.37, "duration": 2.85}, {"text": "I want to modify that\nstrategy so that we", "start": 145.22, "duration": 3.69}, {"text": "can work in the integers.", "start": 148.91, "duration": 2.67}, {"text": "Unlike an F2 to\nthe N, where things", "start": 151.58, "duration": 2.61}, {"text": "were fairly nice and clean,\nbecause you have subspaces,", "start": 154.19, "duration": 3.42}, {"text": "you can take a\nFourier coefficient,", "start": 157.61, "duration": 1.92}, {"text": "pass it down to a subspace.", "start": 159.53, "duration": 2.37}, {"text": "There is no subspaces, right?", "start": 161.9, "duration": 1.86}, {"text": "There are no subspaces\nin the integers.", "start": 163.76, "duration": 3.22}, {"text": "So we have to do something\nslightly different,", "start": 166.98, "duration": 2.04}, {"text": "but in the same spirit.", "start": 169.02, "duration": 2.1}, {"text": "So you'll find a large\nFourier coefficient.", "start": 171.12, "duration": 2.457}, {"text": "And we will find that there\nis density increment when", "start": 180.4, "duration": 9.97}, {"text": "you restrict not to\nsubspaces, but what", "start": 190.37, "duration": 4.04}, {"text": "could play the role of subspaces\nwhen it comes to the integers?", "start": 194.41, "duration": 2.925}, {"text": "So I want something which\nlooks like a smaller", "start": 199.9, "duration": 3.15}, {"text": "version of the original space.", "start": 203.05, "duration": 3.21}, {"text": "So instead of it\nbeing integers, if we", "start": 206.26, "duration": 3.02}, {"text": "restrict to a subprogression,\nso to a smaller arithmetic", "start": 209.28, "duration": 4.5}, {"text": "progression.", "start": 213.78, "duration": 1.19}, {"text": "I will show that you can\nrestrict to a subprogression", "start": 214.97, "duration": 3.07}, {"text": "where you can obtain\ndensity increment.", "start": 218.04, "duration": 2.657}, {"text": "So we'll restrict integers\nto something smaller.", "start": 225.84, "duration": 4.23}, {"text": "And then, same as last time,\nwe can iterate this increment", "start": 230.07, "duration": 14.61}, {"text": "to obtain the\nconclusion that you", "start": 244.68, "duration": 3.81}, {"text": "have an upper bound on the\nsize of this 3AP-free set.", "start": 248.49, "duration": 6.055}, {"text": "OK, so that's the strategy.", "start": 254.545, "duration": 1.125}, {"text": "So you see the same strategy\nas the one we did last time,", "start": 255.67, "duration": 2.64}, {"text": "and many of the ingredients\nwill have parallels,", "start": 258.31, "duration": 2.519}, {"text": "but the execution will\nbe slightly different,", "start": 260.829, "duration": 3.001}, {"text": "especially in the\nsecond step where,", "start": 263.83, "duration": 2.92}, {"text": "because we no longer\nhave sub spaces, which", "start": 266.75, "duration": 2.69}, {"text": "are nice and clean,\nso that's why", "start": 269.44, "duration": 1.44}, {"text": "we started with a\nfinite fuel model, just", "start": 270.88, "duration": 2.19}, {"text": "to show how things work in\na slightly easier setting.", "start": 273.07, "duration": 2.46}, {"text": "And today, we'll see how to\ndo a same kind of strategy", "start": 275.53, "duration": 4.14}, {"text": "here, where there is going\nto be a bit more work.", "start": 279.67, "duration": 3.22}, {"text": "Not too much more,\nbut a bit more work.", "start": 282.89, "duration": 2.4}, {"text": "OK, so before we\nstart, any questions?", "start": 285.29, "duration": 2.501}, {"text": "All right.", "start": 293.2, "duration": 1.72}, {"text": "So last time I used the\nproof Roth's theorem", "start": 294.92, "duration": 2.37}, {"text": "as an excuse to introduce\nFourier analysis.", "start": 297.29, "duration": 1.955}, {"text": "And we're going to see basically\nthe same kind of Fourier", "start": 299.245, "duration": 2.375}, {"text": "analysis, but it's going to take\non a slightly different form,", "start": 301.62, "duration": 3.8}, {"text": "because we're not working\nat F3 to the N. We're", "start": 305.42, "duration": 2.52}, {"text": "working inside the integers.", "start": 307.94, "duration": 2.55}, {"text": "And there's a general theory of\nFourier analysis on the group,", "start": 310.49, "duration": 3.38}, {"text": "on a billion groups.", "start": 313.87, "duration": 2.602}, {"text": "I don't want to go into that\ntheory, because that's--", "start": 316.472, "duration": 2.208}, {"text": "I want to focus on\nthe specific case,", "start": 318.68, "duration": 1.762}, {"text": "but the point is that\ngiven the billion groups,", "start": 320.442, "duration": 1.958}, {"text": "you always have a dual\ngroup of characters.", "start": 322.4, "duration": 4.82}, {"text": "And they play the role\nof Fourier transform.", "start": 327.22, "duration": 2.58}, {"text": "Specifically in\nthe case of Z, we", "start": 329.8, "duration": 3.99}, {"text": "have the following\nFourier transform.", "start": 333.79, "duration": 2.1}, {"text": "So the dual group of Z\nturns out to be the torus.", "start": 341.39, "duration": 9.72}, {"text": "So real numbers mod one.", "start": 351.11, "duration": 3.43}, {"text": "And the Fourier\ntransform is defined", "start": 354.54, "duration": 8.91}, {"text": "as follows, starting with a\nfunction on the integers, OK.", "start": 363.45, "duration": 11.46}, {"text": "If you'd like, let's say\nit's finitely supported,", "start": 374.91, "duration": 2.1}, {"text": "just to make our\nlives a bit easier.", "start": 377.01, "duration": 1.72}, {"text": "Don't have to deal\nwith technicalities.", "start": 378.73, "duration": 1.95}, {"text": "But in general, the\nfollowing formula holds.", "start": 380.68, "duration": 3.14}, {"text": "We have this Fourier\ntransform defined", "start": 383.82, "duration": 8.46}, {"text": "by setting f hat of theta\nto be the following sum.", "start": 392.28, "duration": 7.34}, {"text": "OK, where this e is actually\nsomewhat standard notation,", "start": 404.922, "duration": 5.108}, {"text": "additive combinatorics.", "start": 410.03, "duration": 2.45}, {"text": "It's e to the 2\npi i t, all right?", "start": 412.48, "duration": 3.92}, {"text": "So it goes a fraction, t,\naround the complex unit circle.", "start": 416.4, "duration": 5.95}, {"text": "OK.", "start": 422.35, "duration": 0.5}, {"text": "So that's the Fourier\ntransform on the integers.", "start": 422.85, "duration": 2.657}, {"text": "OK, so you might have seen this\nbefore under a different name.", "start": 425.507, "duration": 2.583}, {"text": "This is usually\ncalled Fourier series.", "start": 428.09, "duration": 1.68}, {"text": "All right.", "start": 433.79, "duration": 0.5}, {"text": "You know, the notation\nmay be slightly different.", "start": 434.29, "duration": 2.042}, {"text": "OK, so that's what\nwe'll see today.", "start": 436.332, "duration": 2.718}, {"text": "And this Fourier transform plays\nthe same role as the Fourier", "start": 439.05, "duration": 3.71}, {"text": "transform from last time, which\nwas on the group F3 to the N.", "start": 442.76, "duration": 6.63}, {"text": "And just us in--", "start": 449.39, "duration": 3.672}, {"text": "so last time, we had a number\nof important identities,", "start": 453.062, "duration": 3.958}, {"text": "and we'll have the same\nkinds of identities here.", "start": 457.02, "duration": 2.88}, {"text": "So let me remind\nyou what they are.", "start": 459.9, "duration": 2.07}, {"text": "And the proofs are all\nbasically the same,", "start": 461.97, "duration": 1.84}, {"text": "so I won't show you the proofs.", "start": 463.81, "duration": 1.292}, {"text": "f hat of 0 is simply the\nsum of f over the domain.", "start": 474.13, "duration": 7.41}, {"text": "We have this Plancherel Parseval\nidentity, which tells us", "start": 485.22, "duration": 11.25}, {"text": "that if you look at\nthe inner product", "start": 496.47, "duration": 6.15}, {"text": "by linear form in\nthe physical space,", "start": 502.62, "duration": 4.99}, {"text": "it equals to the inner\nproduct in the Fourier space.", "start": 507.61, "duration": 5.84}, {"text": "OK.", "start": 522.62, "duration": 0.5}, {"text": "So in the physical\nspace now, you sum.", "start": 523.12, "duration": 2.13}, {"text": "In the frequency space,\nyou take integral", "start": 525.25, "duration": 2.1}, {"text": "over the torus, or the\ncircle, in this case.", "start": 527.35, "duration": 4.61}, {"text": "It's a one-dimensional torus.", "start": 531.96, "duration": 2.61}, {"text": "There is also the Fourier\ninversion formula,", "start": 534.57, "duration": 3.9}, {"text": "which now says that f of x\nis equal to f hat of theta.", "start": 538.47, "duration": 6.88}, {"text": "E of x theta, you integrate\ntheta from 0 to 1.", "start": 545.35, "duration": 4.43}, {"text": "Again, on the torus,\non the circle.", "start": 549.78, "duration": 4.05}, {"text": "And third-- and finally, there\nwas this identity last time", "start": 553.83, "duration": 4.23}, {"text": "that related three-term\narithmetic progressions", "start": 558.06, "duration": 2.88}, {"text": "to the Fourier transform, OK?", "start": 560.94, "duration": 1.74}, {"text": "So this last one was\nslightly not as--", "start": 562.68, "duration": 3.85}, {"text": "I mean, it's not as standard\nas the first several, which are", "start": 566.53, "duration": 3.02}, {"text": "standard Fourier identities.", "start": 569.55, "duration": 2.01}, {"text": "But this one will\nbe useful to us.", "start": 571.56, "duration": 2.8}, {"text": "So the identity relating the\nFourier transform, the 3AP, now", "start": 574.36, "duration": 3.92}, {"text": "has the following form.", "start": 578.28, "duration": 1.363}, {"text": "OK, so if we define\nlambda of f, g,", "start": 579.643, "duration": 7.455}, {"text": "and h to be the\nfollowing sum, which", "start": 587.098, "duration": 8.692}, {"text": "sums over all 3APs\nin the integers,", "start": 595.79, "duration": 6.81}, {"text": "then one can write\nthis expression", "start": 602.6, "duration": 8.87}, {"text": "in terms of the Fourier\ntransform as follows.", "start": 611.47, "duration": 4.5}, {"text": "OK.", "start": 626.75, "duration": 0.97}, {"text": "All right.", "start": 627.72, "duration": 0.5}, {"text": "So comparing this formula to the\none that we saw from last time,", "start": 628.22, "duration": 3.72}, {"text": "it's the same formula,\nwhere different domains,", "start": 631.94, "duration": 2.55}, {"text": "where you're summing\nor integrating,", "start": 634.49, "duration": 1.5}, {"text": "but it's the same formula.", "start": 635.99, "duration": 1.17}, {"text": "And the proof is the same.", "start": 637.16, "duration": 1.84}, {"text": "So go look at the proof.", "start": 639.0, "duration": 1.07}, {"text": "It's the same proof.", "start": 640.07, "duration": 0.833}, {"text": "OK.", "start": 643.71, "duration": 0.5}, {"text": "So these are the key Fourier\nthings that we'll use.", "start": 644.21, "duration": 3.6}, {"text": "And then we'll try to\nfollow on those with--", "start": 647.81, "duration": 3.17}, {"text": "the same as the proof as last\ntime, and see where we can get.", "start": 650.98, "duration": 4.19}, {"text": "So let me introduce\none more notation.", "start": 655.17, "duration": 2.01}, {"text": "So I'll write lambda sub 3\nof f to be lambda of f, f, f,", "start": 657.18, "duration": 7.31}, {"text": "three times.", "start": 664.49, "duration": 0.54}, {"text": "OK.", "start": 667.509, "duration": 0.5}, {"text": "So at this point, if you\nunderstood the lecture", "start": 668.009, "duration": 1.971}, {"text": "from last time, none of\nanything I've said so far", "start": 669.98, "duration": 3.24}, {"text": "should be surprising.", "start": 673.22, "duration": 1.23}, {"text": "We are working\nintegers, so we should", "start": 674.45, "duration": 1.83}, {"text": "look at the corresponding\nFourier transform in integers.", "start": 676.28, "duration": 2.89}, {"text": "And if you follow your\nnotes, this is all the things", "start": 679.17, "duration": 2.51}, {"text": "that we're going to use.", "start": 681.68, "duration": 1.98}, {"text": "OK, so what was one\nof the first things", "start": 683.66, "duration": 2.31}, {"text": "we mentioned regarding\nthe Fourier transform", "start": 685.97, "duration": 3.3}, {"text": "from last time after this point?", "start": 689.27, "duration": 1.56}, {"text": "OK.", "start": 700.25, "duration": 0.5}, {"text": "AUDIENCE: The counting lemma.", "start": 700.75, "duration": 1.325}, {"text": "YUFEI ZHAO: OK.", "start": 702.075, "duration": 0.625}, {"text": "So let's do a counting lemma.", "start": 702.7, "duration": 1.56}, {"text": "So what should the\ncounting lemma say?", "start": 704.26, "duration": 5.52}, {"text": "Well, the spirit of the counting\nlemma is that if you have two", "start": 709.78, "duration": 2.92}, {"text": "functions that are close to\neach other-- and now, \"close\"", "start": 712.7, "duration": 3.66}, {"text": "means close in Fourier--", "start": 716.36, "duration": 2.58}, {"text": "then their corresponding number\nof 3APs should be similar, OK?", "start": 718.94, "duration": 4.97}, {"text": "So that's what we want to say.", "start": 723.91, "duration": 2.24}, {"text": "And indeed, right, so\nthe counting lemma for us", "start": 726.15, "duration": 8.59}, {"text": "will say that if f and g\nare functions on Z, and--", "start": 734.74, "duration": 10.77}, {"text": "such that their L2 norms are\nboth bounded by this M. OK,", "start": 748.42, "duration": 18.955}, {"text": "the sum of the squared absolute\nvalue entries are both bounded.", "start": 767.375, "duration": 4.565}, {"text": "Then the difference\nof their 3AP counts", "start": 771.94, "duration": 11.77}, {"text": "should not be so different\nfrom each other if f", "start": 783.71, "duration": 4.86}, {"text": "and g are close in Fourier, OK?", "start": 788.57, "duration": 2.22}, {"text": "And that means that if all\nthe Fourier coefficients of f", "start": 790.79, "duration": 11.2}, {"text": "minus g are small,\nthen lambda 3ff,", "start": 801.99, "duration": 6.28}, {"text": "which considers 3AP counts\nin f, is close to that of g.", "start": 808.27, "duration": 4.02}, {"text": "OK.", "start": 815.685, "duration": 2.425}, {"text": "Same kind of 3AP counting\nlemma from last time.", "start": 818.11, "duration": 4.634}, {"text": "OK, so let's prove it, OK?", "start": 822.744, "duration": 2.962}, {"text": "As with the counting\nlemma proofs", "start": 831.52, "duration": 2.1}, {"text": "you've seen several times\nalready in this course,", "start": 833.62, "duration": 2.73}, {"text": "we will prove it by first\nwriting this difference", "start": 836.35, "duration": 3.72}, {"text": "as a telescoping sum.", "start": 840.07, "duration": 1.38}, {"text": "The first term\nbeing f minus g f f,", "start": 847.42, "duration": 5.5}, {"text": "and then g of f minus g f and\nlambda g, g, f, f minus g.", "start": 852.92, "duration": 12.125}, {"text": "OK, and we will like to show\nthat each of these terms", "start": 865.045, "duration": 2.745}, {"text": "is small if f minus g has\nsmall Fourier coefficients.", "start": 867.79, "duration": 6.483}, {"text": "OK.", "start": 879.69, "duration": 0.5}, {"text": "So let's bound the first term.", "start": 880.19, "duration": 1.71}, {"text": "OK, so let me bound this first\nterm using the 3AP identity,", "start": 890.077, "duration": 5.263}, {"text": "relating 3AP to\nFourier coefficients,", "start": 895.34, "duration": 2.61}, {"text": "we can write this lambda\nas the following integral", "start": 897.95, "duration": 7.2}, {"text": "over Fourier coefficients.", "start": 905.15, "duration": 1.968}, {"text": "And now, let me--", "start": 924.07, "duration": 1.66}, {"text": "OK, so what was the\ntrick last time?", "start": 925.73, "duration": 1.82}, {"text": "So we said let's pull\nout one of these guys", "start": 927.55, "duration": 5.05}, {"text": "and then use triangle inequality\non the remaining factors.", "start": 932.6, "duration": 4.645}, {"text": "OK, so we'll do that.", "start": 937.245, "duration": 0.905}, {"text": "So far, so good.", "start": 956.41, "duration": 2.442}, {"text": "And now you see this integral.", "start": 958.852, "duration": 3.443}, {"text": "Apply Cauchy-Schwartz.", "start": 962.295, "duration": 1.478}, {"text": "OK, so apply Cauchy-Schwartz\nto the first factor,", "start": 978.09, "duration": 2.8}, {"text": "you got this l2 sum,\nthis l2 integral.", "start": 980.89, "duration": 4.643}, {"text": "And then you apply\nCauchy-Schwartz", "start": 985.533, "duration": 1.417}, {"text": "to the second factor.", "start": 986.95, "duration": 1.345}, {"text": "You get that integral.", "start": 994.633, "duration": 0.917}, {"text": "OK, now what do we do?", "start": 998.826, "duration": 2.094}, {"text": "Yep?", "start": 1004.912, "duration": 1.497}, {"text": "AUDIENCE: [INAUDIBLE]", "start": 1006.409, "duration": 3.571}, {"text": "YUFEI ZHAO: OK, so yeah.", "start": 1009.98, "duration": 1.0}, {"text": "So you see an l2 of Fourier,\nthe first automatic reaction", "start": 1010.98, "duration": 3.72}, {"text": "should be to use a\nPlancherel or Parseval, OK?", "start": 1014.7, "duration": 2.19}, {"text": "So apply Plancherel identity\nto each of these factors.", "start": 1016.89, "duration": 7.699}, {"text": "We find that each\nof those factors", "start": 1024.589, "duration": 8.931}, {"text": "is equal to this l2 sum\nin the physical space.", "start": 1033.52, "duration": 6.52}, {"text": "OK, so this square\nroot, the same thing.", "start": 1043.75, "duration": 1.859}, {"text": "Square root again.", "start": 1045.609, "duration": 2.371}, {"text": "OK.", "start": 1047.98, "duration": 0.54}, {"text": "And then we find\nthat because there", "start": 1048.52, "duration": 6.23}, {"text": "was an hypothesis--\nin the hypothesis,", "start": 1054.75, "duration": 1.97}, {"text": "there was a bound M on\nthis sum of squares.", "start": 1056.72, "duration": 3.71}, {"text": "You have that down there.", "start": 1063.61, "duration": 2.48}, {"text": "And similarly, with\nthe other two terms.", "start": 1066.09, "duration": 1.99}, {"text": "OK.", "start": 1081.526, "duration": 0.524}, {"text": "So that proves the\ncounting lemma.", "start": 1082.05, "duration": 1.461}, {"text": "Question?", "start": 1087.359, "duration": 1.443}, {"text": "AUDIENCE: Last time, the\nterm on the right-hand side", "start": 1088.802, "duration": 3.958}, {"text": "was the maximum over\nnon-zero frequency?", "start": 1092.76, "duration": 3.765}, {"text": "YUFEI ZHAO: OK.", "start": 1096.525, "duration": 1.205}, {"text": "OK.", "start": 1097.73, "duration": 0.5}, {"text": "So the question is, last time\nwe had a counting lemma that", "start": 1098.23, "duration": 4.0}, {"text": "looked slightly different.", "start": 1102.23, "duration": 1.89}, {"text": "But I claimed they're all\nreally the same counting lemma.", "start": 1104.12, "duration": 2.61}, {"text": "They're all the same proofs.", "start": 1106.73, "duration": 1.59}, {"text": "If you run this\nproof, it won't work.", "start": 1108.32, "duration": 2.49}, {"text": "If you take what\nwe did last time,", "start": 1110.81, "duration": 2.34}, {"text": "it's the same kind of proofs.", "start": 1113.15, "duration": 2.11}, {"text": "So last time we had\na counting lemma", "start": 1115.26, "duration": 1.5}, {"text": "where we had the same\nf, f, f essentially.", "start": 1116.76, "duration": 7.36}, {"text": "We know have-- I allow you\nto essentially take three", "start": 1124.12, "duration": 4.54}, {"text": "different things, and--", "start": 1128.66, "duration": 0.98}, {"text": "OK, so both-- in\nboth cases, you're", "start": 1129.64, "duration": 3.34}, {"text": "running through\nthis calculation,", "start": 1132.98, "duration": 1.77}, {"text": "but they look\nslightly different.", "start": 1134.75, "duration": 2.628}, {"text": "AUDIENCE: [INAUDIBLE]", "start": 1137.378, "duration": 5.177}, {"text": "YUFEI ZHAO: So, yeah.", "start": 1142.555, "duration": 0.875}, {"text": "So I agree.", "start": 1143.43, "duration": 1.005}, {"text": "It doesn't look\nexactly the same,", "start": 1144.435, "duration": 1.375}, {"text": "but if you think about\nwhat's involved in the proof,", "start": 1145.81, "duration": 1.71}, {"text": "they're the same proofs.", "start": 1147.52, "duration": 1.0}, {"text": "OK.", "start": 1151.232, "duration": 1.398}, {"text": "Any more questions?", "start": 1152.63, "duration": 3.54}, {"text": "All right.", "start": 1156.17, "duration": 2.89}, {"text": "So now, we have\nthis counting lemma,", "start": 1159.06, "duration": 1.61}, {"text": "so let's start our proof of\nRoth's theorem in the integers.", "start": 1160.67, "duration": 4.99}, {"text": "As with last time, there\nwill be three steps,", "start": 1165.66, "duration": 2.16}, {"text": "as mentioned up there, OK?", "start": 1167.82, "duration": 4.355}, {"text": "In the first step, let us\nshow that if you are 3AP-free,", "start": 1172.175, "duration": 5.415}, {"text": "then we can obtain a density--", "start": 1177.59, "duration": 2.54}, {"text": "a large Fourier coefficient.", "start": 1180.13, "duration": 2.068}, {"text": "Yeah, so in this course,\nthis counting lemma,", "start": 1196.64, "duration": 2.56}, {"text": "we actually solved this--", "start": 1199.2, "duration": 2.25}, {"text": "basically this kind of proof\nfor the first time when we", "start": 1201.45, "duration": 2.52}, {"text": "discussed graph counting\nlemma, back in the chapter", "start": 1203.97, "duration": 3.03}, {"text": "on Szemer\u00e9di's regularity lemma.", "start": 1207.0, "duration": 2.81}, {"text": "And sure, they all\nlook literally--", "start": 1209.81, "duration": 2.62}, {"text": "not exactly the same,\nbut they're all really", "start": 1212.43, "duration": 2.31}, {"text": "the same kind of proofs, right?", "start": 1214.74, "duration": 1.66}, {"text": "So I want--", "start": 1216.4, "duration": 0.65}, {"text": "I'm showing you the same thing\nin many different guises.", "start": 1217.05, "duration": 2.49}, {"text": "But they're all the same proofs.", "start": 1219.54, "duration": 1.333}, {"text": "So if you are a set\nthat is 3AP-free--", "start": 1223.69, "duration": 4.85}, {"text": "and as with last time,\nI'm going to call", "start": 1237.63, "duration": 3.85}, {"text": "alpha the density of A now\ninside this progression,", "start": 1241.48, "duration": 5.07}, {"text": "this length N progression.", "start": 1246.55, "duration": 3.27}, {"text": "And suppose N is large enough.", "start": 1249.82, "duration": 4.74}, {"text": "OK, so the conclusion now is\nthat there exists some theta", "start": 1259.46, "duration": 8.12}, {"text": "such that if you look\nat this sum over here", "start": 1267.58, "duration": 11.59}, {"text": "as a sum over both integers--", "start": 1279.17, "duration": 4.76}, {"text": "actually, let me do\nthe sum only from 1", "start": 1283.93, "duration": 2.75}, {"text": "to uppercase N. Claim that--", "start": 1286.68, "duration": 8.36}, {"text": "OK, so-- so it's saying\nwhat this title says.", "start": 1295.04, "duration": 4.47}, {"text": "If you are 3AP-free\nand this N is large", "start": 1299.51, "duration": 3.695}, {"text": "enough relative to the density,\nyou think of this density", "start": 1303.205, "duration": 2.375}, {"text": "alpha is a constant, then\nI can find a large Fourier", "start": 1305.58, "duration": 4.41}, {"text": "coefficient.", "start": 1309.99, "duration": 0.87}, {"text": "Now, there's a small\ndifference, and this", "start": 1310.86, "duration": 3.15}, {"text": "is related to what you\nwere asking earlier,", "start": 1314.01, "duration": 2.13}, {"text": "between how we set things up now\nversus what happened last time.", "start": 1316.14, "duration": 3.57}, {"text": "So last time, we just looked\nfor a Fourier coefficient", "start": 1319.71, "duration": 5.82}, {"text": "corresponding to a non-zero r.", "start": 1325.53, "duration": 4.55}, {"text": "Now, I'm not\nrestricting non-zero,", "start": 1330.08, "duration": 3.26}, {"text": "but I don't start with\nan indicator function.", "start": 1333.34, "duration": 2.36}, {"text": "I start with the demeaned\nindicator function.", "start": 1335.7, "duration": 3.58}, {"text": "I take out the mean so that\nthe zeroth coefficient,", "start": 1339.28, "duration": 5.06}, {"text": "so to speak, which corresponds\nto the mean, is already 0.", "start": 1344.34, "duration": 4.11}, {"text": "So you don't get to use\nthat for your coefficient.", "start": 1348.45, "duration": 4.34}, {"text": "So if you didn't do\nthis, if you just", "start": 1352.79, "duration": 2.05}, {"text": "tried to do this\nlast time, I mean,", "start": 1354.84, "duration": 1.58}, {"text": "you can also do\nexactly the same setup.", "start": 1356.42, "duration": 1.625}, {"text": "But if you don't\ndemean it, then--", "start": 1358.045, "duration": 2.135}, {"text": "if you don't have this\nterm, then this statement", "start": 1360.18, "duration": 2.22}, {"text": "is trivially true, because I\ncan take theta equal to 0, OK?", "start": 1362.4, "duration": 4.56}, {"text": "But I don't want.", "start": 1366.96, "duration": 1.34}, {"text": "I want an actual significant\nFourier improvement.", "start": 1368.3, "duration": 4.6}, {"text": "So I take--", "start": 1372.9, "duration": 1.03}, {"text": "I do this demean, and then\nI consider its Fourier", "start": 1373.93, "duration": 4.61}, {"text": "coefficient.", "start": 1378.54, "duration": 1.043}, {"text": "OK.", "start": 1382.26, "duration": 0.5}, {"text": "Any questions about\nthe statement?", "start": 1382.76, "duration": 3.46}, {"text": "Yeah, so this demeaning is\nreally important, right?", "start": 1386.22, "duration": 3.59}, {"text": "So that's something that's a\nvery common technique whenever", "start": 1389.81, "duration": 2.72}, {"text": "you do these kind of analysis.", "start": 1392.53, "duration": 1.61}, {"text": "So make sure you're--", "start": 1394.14, "duration": 2.37}, {"text": "so that you're-- yeah, so you're\nlooking at functions with mean", "start": 1396.51, "duration": 3.43}, {"text": "0.", "start": 1399.94, "duration": 2.837}, {"text": "Let's see the proof.", "start": 1402.777, "duration": 0.833}, {"text": "We have the\nfollowing information", "start": 1408.19, "duration": 5.1}, {"text": "about all 3AP counts in A.\nBecause A is 3AP-free, OK,", "start": 1413.29, "duration": 6.38}, {"text": "so what is the value of lambda\nsub 3 of the indicator of A?", "start": 1419.67, "duration": 3.65}, {"text": "Lambda of 3, if you\nlook at the expression,", "start": 1426.1, "duration": 3.34}, {"text": "it basically sums over all\n3APs, but A has no 3APs,", "start": 1429.44, "duration": 6.33}, {"text": "except for the trivial ones.", "start": 1435.77, "duration": 2.73}, {"text": "So we'll only consider\nthe trivial 3APs,", "start": 1438.5, "duration": 2.52}, {"text": "which has size exactly\nthe size of A, which", "start": 1441.02, "duration": 5.19}, {"text": "is alpha N from trivial 3APs.", "start": 1446.21, "duration": 4.61}, {"text": "On the other hand, what\ndo we know about lambda 3", "start": 1454.87, "duration": 5.16}, {"text": "of this interval from 1 to N?", "start": 1460.03, "duration": 5.226}, {"text": "OK, so how many 3APs are there?", "start": 1465.256, "duration": 3.154}, {"text": "OK, so roughly, it's going\nto be about N squared over 2.", "start": 1468.41, "duration": 5.23}, {"text": "And in fact, it will be\nat least N squared over 2,", "start": 1473.64, "duration": 4.83}, {"text": "because to generate\na 3AP, I just", "start": 1478.47, "duration": 3.39}, {"text": "have to pick a first\nterm and a third term,", "start": 1481.86, "duration": 7.21}, {"text": "and I'm OK as long as\nthey're the same parity.", "start": 1489.07, "duration": 2.69}, {"text": "And then you have a 3AP.", "start": 1498.195, "duration": 2.215}, {"text": "So the same parity\ncuts you down by half,", "start": 1500.41, "duration": 3.81}, {"text": "so you have at least N squared\nover 2 3APs from 1 through N.", "start": 1504.22, "duration": 6.97}, {"text": "So now, let's look at\nhow to apply the counting", "start": 1511.19, "duration": 3.74}, {"text": "lemma all to the setting.", "start": 1514.93, "duration": 1.55}, {"text": "So we have the counting\nlemma up there,", "start": 1516.48, "duration": 1.94}, {"text": "where I now want to apply it--", "start": 1518.42, "duration": 1.51}, {"text": "so apply counting\nto, on one hand,", "start": 1522.6, "duration": 6.36}, {"text": "the indicator function of A\nso we get the count 3APs in A,", "start": 1528.96, "duration": 5.82}, {"text": "but also compared to\nthe normalized indicator", "start": 1534.78, "duration": 8.99}, {"text": "on the interval.", "start": 1543.77, "duration": 1.482}, {"text": "OK, so maybe this is\na good point for me", "start": 1545.252, "duration": 1.688}, {"text": "to pause and remind you that\nthe spirit of this whole proof", "start": 1546.94, "duration": 3.485}, {"text": "is understanding structure\nversus pseudorandomness, OK?", "start": 1550.425, "duration": 4.525}, {"text": "So as was the case last time.", "start": 1554.95, "duration": 2.73}, {"text": "So we want to understand, in\nwhat ways is A pseudorandom?", "start": 1557.68, "duration": 5.94}, {"text": "And here, \"pseudorandom,\"\njust as with last time,", "start": 1563.62, "duration": 2.4}, {"text": "means having small\nFourier coefficients,", "start": 1566.02, "duration": 2.52}, {"text": "being Fourier uniform.", "start": 1568.54, "duration": 3.316}, {"text": "If A is pseudorandom,\nwhich here, means f and g", "start": 1571.856, "duration": 4.994}, {"text": "are close to each other.", "start": 1576.85, "duration": 1.98}, {"text": "That's what being\npseudorandom means,", "start": 1578.83, "duration": 1.59}, {"text": "then the counting\nlemma will tell us", "start": 1580.42, "duration": 1.5}, {"text": "that f and g should\nhave similar AP counts.", "start": 1581.92, "duration": 3.87}, {"text": "But A has basically no AP\ncount, so they should not", "start": 1585.79, "duration": 5.4}, {"text": "be close to each other.", "start": 1591.19, "duration": 4.78}, {"text": "So that's the strategy, to\nshow that A is not pseudorandom", "start": 1595.97, "duration": 2.64}, {"text": "in this sense, and thereby\nextracting a large Fourier", "start": 1598.61, "duration": 2.88}, {"text": "coefficient.", "start": 1601.49, "duration": 1.75}, {"text": "So we apply counting\nto these two functions,", "start": 1603.24, "duration": 4.25}, {"text": "and we obtain that.", "start": 1607.49, "duration": 1.86}, {"text": "OK.", "start": 1614.86, "duration": 0.5}, {"text": "So this quantity, which\ncorresponds to lambda 3 of g,", "start": 1615.36, "duration": 7.64}, {"text": "minus alpha N. So these were\nlambda 3 of g, lambda 3 of F.", "start": 1623.0, "duration": 14.3}, {"text": "So it is upper-bounded.", "start": 1637.3, "duration": 1.46}, {"text": "The difference is up\nrebounded by the--", "start": 1638.76, "duration": 5.137}, {"text": "using the counting\nlemma, we find", "start": 1648.01, "duration": 1.75}, {"text": "that their difference\nis upper-bounded", "start": 1649.76, "duration": 1.74}, {"text": "by the following quantity.", "start": 1651.5, "duration": 1.17}, {"text": "Namely, you look at the\ndifference between f and g", "start": 1663.76, "duration": 3.09}, {"text": "and evaluate its maximum\nFourier coefficient.", "start": 1666.85, "duration": 5.232}, {"text": "OK.", "start": 1672.082, "duration": 0.928}, {"text": "So if A is pseudorandom,\nmeaning that Fourier uniform--", "start": 1673.01, "duration": 5.75}, {"text": "this l infinity\nnorm is small, then", "start": 1678.76, "duration": 6.08}, {"text": "I should expect lots\nand lots of 3APs in A,", "start": 1684.84, "duration": 5.17}, {"text": "but because that\nis not the case,", "start": 1690.01, "duration": 2.38}, {"text": "we should be able to\nconclude that there is", "start": 1692.39, "duration": 2.12}, {"text": "some large Fourier coefficient.", "start": 1694.51, "duration": 2.235}, {"text": "All right, so thus--", "start": 1700.0, "duration": 1.09}, {"text": "so rearranging the equation\nabove, we have that--", "start": 1713.131, "duration": 3.589}, {"text": "so this should be a square.", "start": 1724.03, "duration": 1.24}, {"text": "OK.", "start": 1731.48, "duration": 0.62}, {"text": "So we have this expression here.", "start": 1732.1, "duration": 2.16}, {"text": "And now we are--", "start": 1734.26, "duration": 2.43}, {"text": "OK, so let me simplify\nthis expression slightly.", "start": 1736.69, "duration": 6.63}, {"text": "And now we're using that N\nis sufficiently large, OK?", "start": 1743.32, "duration": 4.23}, {"text": "So we're using N is\nsufficiently large.", "start": 1747.55, "duration": 4.74}, {"text": "So this quantity is at least\na tenth of alpha squared N.", "start": 1752.29, "duration": 10.912}, {"text": "OK, and that's the\nconclusion, all right?", "start": 1763.202, "duration": 1.708}, {"text": "So that's the conclusion\nof this step here.", "start": 1764.91, "duration": 4.027}, {"text": "What does this mean?", "start": 1768.937, "duration": 0.833}, {"text": "This means there\nexists some theta", "start": 1769.77, "duration": 3.83}, {"text": "so that the Fourier\ncoefficient at theta", "start": 1773.6, "duration": 3.45}, {"text": "is at least the\nclaimed quantity.", "start": 1777.05, "duration": 1.7}, {"text": "Any questions?", "start": 1782.53, "duration": 1.304}, {"text": "All right.", "start": 1791.42, "duration": 0.63}, {"text": "So that finishes step 1.", "start": 1792.05, "duration": 1.91}, {"text": "So now let me go on step 2.", "start": 1793.96, "duration": 3.76}, {"text": "In step 2, we wish to show that\nif you have a large Fourier", "start": 1797.72, "duration": 4.4}, {"text": "coefficient, then one can\nobtain a density increment.", "start": 1802.12, "duration": 9.836}, {"text": "So last time, we were working\nin a finite field vector space.", "start": 1821.2, "duration": 5.52}, {"text": "A Fourier coefficient, OK,\nso which is a dual vector,", "start": 1826.72, "duration": 3.75}, {"text": "corresponds to some hyperplane.", "start": 1830.47, "duration": 3.55}, {"text": "And having a large\nFourier coefficient", "start": 1834.02, "duration": 3.59}, {"text": "then implies that\nthe density of A", "start": 1837.61, "duration": 2.77}, {"text": "on the co-sets of\nthose hyperplanes", "start": 1840.38, "duration": 2.76}, {"text": "must be not all\nclose to each other.", "start": 1843.14, "duration": 3.58}, {"text": "All right, so one\nof the hyperplanes", "start": 1846.72, "duration": 1.85}, {"text": "must have significantly\nhigher density than the rest.", "start": 1848.57, "duration": 4.81}, {"text": "OK, so we want to do\nsomething similar here,", "start": 1853.38, "duration": 1.92}, {"text": "except we run into this\ntechnical difficulty", "start": 1855.3, "duration": 2.28}, {"text": "where there are no\nsubspaces anymore.", "start": 1857.58, "duration": 3.67}, {"text": "So the Fourier character, namely\ncorresponding to this theta,", "start": 1861.25, "duration": 3.993}, {"text": "is just a real number.", "start": 1865.243, "duration": 0.917}, {"text": "It doesn't divide up your space.", "start": 1866.16, "duration": 1.35}, {"text": "It doesn't divide\nup your 1 through N", "start": 1867.51, "duration": 1.56}, {"text": "very nicely into sub chunks.", "start": 1869.07, "duration": 1.98}, {"text": "But we still want to use this\ntheta to chop up 1 through N", "start": 1871.05, "duration": 4.77}, {"text": "into smaller spaces so\nthat we can iterate and do", "start": 1875.82, "duration": 5.04}, {"text": "density increment.", "start": 1880.86, "duration": 3.87}, {"text": "All right.", "start": 1884.73, "duration": 1.18}, {"text": "So let's see what we can do.", "start": 1885.91, "duration": 3.01}, {"text": "So given this theta,\nwhat we would like to do", "start": 1888.92, "duration": 8.06}, {"text": "is to partition this 1 through\nN into subprogressions.", "start": 1896.98, "duration": 13.28}, {"text": "OK, so chop up 1 through\nN into sub APs such that", "start": 1913.564, "duration": 13.466}, {"text": "if you evaluate for--\nso this theta is fixed.", "start": 1927.03, "duration": 5.2}, {"text": "So on each sub AP,\nthis function here", "start": 1932.23, "duration": 5.76}, {"text": "is roughly constant\non each of your parts.", "start": 1937.99, "duration": 10.01}, {"text": "Last time, we had this\nFourier character,", "start": 1951.15, "duration": 2.85}, {"text": "and then we chopped it up\nusing these three hyperplanes.", "start": 1954.0, "duration": 3.12}, {"text": "And each hyperplane,\nthe Fourier character", "start": 1957.12, "duration": 3.27}, {"text": "is literally constant, OK?", "start": 1960.39, "duration": 2.688}, {"text": "So you have-- and so\nthat's what we work with.", "start": 1963.078, "duration": 3.592}, {"text": "And now, you cannot get\nthem to be exactly constant,", "start": 1966.67, "duration": 2.78}, {"text": "but the next best thing we can\nhope for is to get this Fourier", "start": 1969.45, "duration": 3.57}, {"text": "character to be\nroughly constant.", "start": 1973.02, "duration": 3.077}, {"text": "OK, so we're going to do some\npositioning that allows us", "start": 1976.097, "duration": 2.333}, {"text": "to achieve this characteristic.", "start": 1978.43, "duration": 2.43}, {"text": "And let me give\nyou some intuition", "start": 1980.86, "duration": 2.88}, {"text": "about why this is true.", "start": 1983.74, "duration": 1.17}, {"text": "And this is not exactly\na surprising fact.", "start": 1984.91, "duration": 2.66}, {"text": "The intuition is\njust that if you", "start": 1987.57, "duration": 2.47}, {"text": "look at what this\nfunction behaves like--", "start": 1990.04, "duration": 1.86}, {"text": "all right, so what's\ngoing on here?", "start": 1997.444, "duration": 2.506}, {"text": "You are on the unit circle,\nand you are jumping by theta.", "start": 1999.95, "duration": 7.65}, {"text": "OK, so you just keep\njumping by theta and so on.", "start": 2011.288, "duration": 10.792}, {"text": "And I want to show that I\ncan sharp up my progression", "start": 2022.08, "duration": 7.29}, {"text": "into a bunch of almost periodic\npieces, where in each part,", "start": 2029.37, "duration": 8.32}, {"text": "I'm staying inside a small arc.", "start": 2037.69, "duration": 3.68}, {"text": "So in the extreme case of this\nwhere it is very easy to see", "start": 2041.37, "duration": 5.19}, {"text": "is if x is some rational number,\na over b, with b fairly small,", "start": 2046.56, "duration": 11.599}, {"text": "then we can--", "start": 2058.159, "duration": 3.851}, {"text": "so then, this character is\nactually constant on APs", "start": 2062.01, "duration": 12.18}, {"text": "with common difference b.", "start": 2074.19, "duration": 1.469}, {"text": "Yep?", "start": 2083.063, "duration": 0.5}, {"text": "AUDIENCE: Is theta\nsupposed to be [INAUDIBLE]??", "start": 2083.563, "duration": 3.145}, {"text": "YUFEI ZHAO: Ah, so theta,\nyes. so theta-- thank you.", "start": 2086.708, "duration": 2.167}, {"text": "So theta 2 pi.", "start": 2088.875, "duration": 1.395}, {"text": "AUDIENCE: Like, is x\nequal to your theta?", "start": 2094.174, "duration": 3.196}, {"text": "YUFEI ZHAO: Yeah.", "start": 2097.37, "duration": 0.708}, {"text": "Thank you.", "start": 2098.078, "duration": 0.976}, {"text": "So theta equals-- yeah.", "start": 2099.054, "duration": 1.466}, {"text": "So if theta is some rational\nwith some small denominator--", "start": 2100.52, "duration": 5.66}, {"text": "so then you are literally\njumping in periodic steps", "start": 2106.18, "duration": 7.24}, {"text": "on the unit circle.", "start": 2113.42, "duration": 2.27}, {"text": "So if you partition N according\nto the exact same periods,", "start": 2115.69, "duration": 5.54}, {"text": "you have that this character\nis exactly constant in each", "start": 2121.23, "duration": 5.07}, {"text": "of your progressions.", "start": 2126.3, "duration": 1.89}, {"text": "Now, in general, the theta\nyou get out of that proof", "start": 2128.19, "duration": 4.02}, {"text": "might not have this\nvery nice form,", "start": 2132.21, "duration": 3.19}, {"text": "but we can at\nleast approximately", "start": 2135.4, "duration": 2.94}, {"text": "achieve the desired effect.", "start": 2138.34, "duration": 1.5}, {"text": "OK.", "start": 2142.63, "duration": 0.5}, {"text": "Any questions?", "start": 2146.89, "duration": 0.81}, {"text": "OK.", "start": 2175.56, "duration": 0.7}, {"text": "So to achieve approximately the\ndesired effect, what we'll do", "start": 2176.26, "duration": 3.84}, {"text": "is to find something\nso that b times theta", "start": 2180.1, "duration": 5.27}, {"text": "is not quite an integer, but\nvery close to an integer.", "start": 2185.37, "duration": 4.115}, {"text": "OK, so this, probably many\nof you have seen before.", "start": 2189.485, "duration": 2.125}, {"text": "It's a classic\npigeonhole-type result.", "start": 2191.61, "duration": 5.86}, {"text": "It's usually attributed\nto Dirichlet.", "start": 2197.47, "duration": 4.04}, {"text": "So if you have theta, a\nreal number, and a delta,", "start": 2205.02, "duration": 9.97}, {"text": "kind of a tolerance,\nthen there exists", "start": 2214.99, "duration": 5.31}, {"text": "a positive integer d at\nmost 1 over delta such", "start": 2220.3, "duration": 9.05}, {"text": "that d times theta is\nvery close to an integer.", "start": 2229.35, "duration": 9.68}, {"text": "OK, so this norm\nhere is distance", "start": 2239.03, "duration": 4.69}, {"text": "to the closest integer.", "start": 2243.72, "duration": 2.13}, {"text": "All right, so the proof is\nby pigeonhole principle.", "start": 2253.215, "duration": 5.175}, {"text": "So if we let N be 1\nover delta rounded down", "start": 2258.39, "duration": 6.17}, {"text": "and consider the numbers 0,\ntheta, 2 theta, 3 theta, and so", "start": 2264.56, "duration": 7.41}, {"text": "on, to N theta--", "start": 2271.97, "duration": 3.66}, {"text": "so by pigeonhole, there\nexists i theta and j theta, so", "start": 2275.63, "duration": 12.41}, {"text": "two different terms\nof the sequence such", "start": 2288.04, "duration": 4.62}, {"text": "that they differ by\nless than-- at most", "start": 2292.66, "duration": 8.04}, {"text": "delta in their fractional parts.", "start": 2300.7, "duration": 1.89}, {"text": "OK, so now take d to be\ndifference between i and j.", "start": 2310.26, "duration": 6.66}, {"text": "OK, and that works.", "start": 2316.92, "duration": 0.9}, {"text": "OK.", "start": 2328.06, "duration": 0.5}, {"text": "So even though you don't\nhave exactly rational,", "start": 2328.56, "duration": 4.16}, {"text": "you have approximately rational.", "start": 2332.72, "duration": 2.49}, {"text": "So this is a--", "start": 2335.21, "duration": 0.99}, {"text": "it's a simple rational\napproximation statement.", "start": 2336.2, "duration": 2.77}, {"text": "And using this\nrational approximation,", "start": 2338.97, "duration": 1.92}, {"text": "we can now try to do\nthe intuition here,", "start": 2340.89, "duration": 4.77}, {"text": "pretending that we're working\nwith rational numbers, indeed.", "start": 2345.66, "duration": 4.83}, {"text": "OK, so if we take\neta between 0 and 1", "start": 2356.442, "duration": 4.228}, {"text": "and theta irrational and\nsuppose N is large enough--", "start": 2360.67, "duration": 14.18}, {"text": "OK, so here, C\nmeans there exists", "start": 2374.85, "duration": 2.24}, {"text": "some sufficiently large--", "start": 2377.09, "duration": 3.4}, {"text": "some constant C such that\nthe statement is true, OK?", "start": 2380.49, "duration": 4.93}, {"text": "So suppose you think\na million here.", "start": 2385.42, "duration": 4.0}, {"text": "That should be fine.", "start": 2389.42, "duration": 1.74}, {"text": "So then there exists--", "start": 2391.16, "duration": 4.58}, {"text": "so then one can partition\n1 through N into sub-APs,", "start": 2395.74, "duration": 16.59}, {"text": "which we'll call P i.", "start": 2412.33, "duration": 2.63}, {"text": "And each having length\nbetween cube root of N", "start": 2414.96, "duration": 9.29}, {"text": "and twice the cube root of\nN such that this character", "start": 2424.25, "duration": 8.86}, {"text": "that we want to stay\nroughly constant indeed", "start": 2433.11, "duration": 2.59}, {"text": "does not change very much.", "start": 2435.7, "duration": 3.85}, {"text": "If you look at two terms in\nthe same AP, in the sub-AP,", "start": 2439.55, "duration": 8.06}, {"text": "then the value of this\ncharacter on each P sub i", "start": 2447.61, "duration": 13.06}, {"text": "is roughly the same.", "start": 2460.67, "duration": 2.41}, {"text": "So they don't vary by\nmore than eta on each P i.", "start": 2463.08, "duration": 5.84}, {"text": "So here, we're partitioning this\n1 through N into a sub-A piece", "start": 2468.92, "duration": 7.43}, {"text": "so that this guy here\nstays roughly constant.", "start": 2476.35, "duration": 4.005}, {"text": "OK.", "start": 2485.94, "duration": 0.5}, {"text": "Any questions?", "start": 2486.44, "duration": 3.03}, {"text": "All right.", "start": 2489.47, "duration": 0.5}, {"text": "So think about how\nyou might prove this.", "start": 2489.97, "duration": 3.758}, {"text": "Let's take a quick break.", "start": 2493.728, "duration": 1.042}, {"text": "So you see, we are basically\nfollowing the same strategy", "start": 2497.58, "duration": 3.58}, {"text": "as the proof from last\ntime, but this second step,", "start": 2501.16, "duration": 3.61}, {"text": "which we're on right now,\nneeds to be somewhat modified", "start": 2504.77, "duration": 2.6}, {"text": "because you cannot cut this\nspace up into pieces where", "start": 2507.37, "duration": 4.86}, {"text": "your character is constant.", "start": 2512.23, "duration": 2.1}, {"text": "Well, if they're roughly\nconstant then we're go to go,", "start": 2514.33, "duration": 3.81}, {"text": "so that's what we're doing now.", "start": 2518.14, "duration": 3.16}, {"text": "So let's prove the\nstatement up there.", "start": 2521.3, "duration": 1.7}, {"text": "All right.", "start": 2535.35, "duration": 0.51}, {"text": "So let's prove this\nstatement over here.", "start": 2535.86, "duration": 3.09}, {"text": "So using Dirichlet's lemma, we\nfind that there exists some d.", "start": 2538.95, "duration": 18.65}, {"text": "OK, so I'll write down\nsome number for now.", "start": 2557.6, "duration": 2.015}, {"text": "Don't worry about it.", "start": 2559.615, "duration": 0.875}, {"text": "It will come up shortly why I\nwrite this specific quantity.", "start": 2560.49, "duration": 4.6}, {"text": "So there exists some d which is\nnot too big, such that d theta", "start": 2567.67, "duration": 18.66}, {"text": "is very close to an integer.", "start": 2586.33, "duration": 3.5}, {"text": "So now, I'm literally\napplying Dirichlet's lemma.", "start": 2589.83, "duration": 2.71}, {"text": "OK.", "start": 2596.05, "duration": 1.63}, {"text": "So given such d--", "start": 2597.68, "duration": 7.146}, {"text": "so how big is this d?", "start": 2604.826, "duration": 2.814}, {"text": "You see that because I assumed\nthat N is sufficiently large,", "start": 2607.64, "duration": 5.1}, {"text": "if we choose that C large\nenough, d is at most root N. So", "start": 2612.74, "duration": 16.06}, {"text": "given such d, which\nis at most root N,", "start": 2628.8, "duration": 4.77}, {"text": "you can partition 1 through\nN into subprogressions", "start": 2633.57, "duration": 10.61}, {"text": "with common difference d.", "start": 2644.18, "duration": 5.4}, {"text": "Essentially, look at--\nlet's do classes mod d.", "start": 2649.58, "duration": 4.97}, {"text": "So they're all going to have\nlength by basically N over d.", "start": 2654.55, "duration": 4.58}, {"text": "And I chop them up a\nlittle bit further to get--", "start": 2659.13, "duration": 6.01}, {"text": "so a piece of length\nbetween cube root of N", "start": 2665.14, "duration": 8.25}, {"text": "and twice cube root of N.", "start": 2673.39, "duration": 7.6}, {"text": "OK.", "start": 2680.99, "duration": 0.5}, {"text": "So I'm going to make\nsure that all of my APs", "start": 2681.49, "duration": 3.03}, {"text": "are roughly the same length.", "start": 2684.52, "duration": 3.48}, {"text": "And now, inside each\nsubprogression--", "start": 2688.0, "duration": 8.09}, {"text": "let me call this subprogression\nP prime, subprogression P,", "start": 2700.79, "duration": 5.82}, {"text": "let's look at how much\nthis character value can", "start": 2706.61, "duration": 2.82}, {"text": "vary inside this progression.", "start": 2709.43, "duration": 3.689}, {"text": "All right.", "start": 2719.346, "duration": 0.958}, {"text": "OK, so how much can this vary?", "start": 2726.01, "duration": 3.41}, {"text": "Well, because theta is such\nthat d times theta is very close", "start": 2729.42, "duration": 6.91}, {"text": "to an integer and the length\nof each progression is not too", "start": 2736.33, "duration": 5.448}, {"text": "large-- so here's--", "start": 2741.778, "duration": 0.792}, {"text": "I want some control\non the length.", "start": 2742.57, "duration": 2.14}, {"text": "So we find that the\nmaximum variation", "start": 2744.71, "duration": 4.31}, {"text": "is, at most, the size of\nP, the length of P, times--", "start": 2749.02, "duration": 7.675}, {"text": "so this-- that\ndifference over there.", "start": 2759.58, "duration": 4.95}, {"text": "So all of these are exponential,\nso I can shift them.", "start": 2764.53, "duration": 4.39}, {"text": "Well, the length of P is at\nmost twice cube root of N. And--", "start": 2768.92, "duration": 11.075}, {"text": "OK, so what is this quantity?", "start": 2779.995, "duration": 2.165}, {"text": "So the point is that if\nthis fractional part here", "start": 2782.16, "duration": 3.31}, {"text": "is very close to an\ninteger, then e to that,", "start": 2785.47, "duration": 3.61}, {"text": "e to the 2 pi times that\ni times some number should", "start": 2789.08, "duration": 2.48}, {"text": "be very close to 1, because\nwhat is happening here?", "start": 2791.56, "duration": 5.07}, {"text": "This is the distance\nbetween those two points", "start": 2796.63, "duration": 5.42}, {"text": "on the circle, which\nis at most bounded", "start": 2802.05, "duration": 3.18}, {"text": "by the length of the arc.", "start": 2805.23, "duration": 1.26}, {"text": "OK, so cord length,\nat most of the arc.", "start": 2817.46, "duration": 5.07}, {"text": "So now, you put\neverything here together,", "start": 2822.53, "duration": 3.03}, {"text": "and apply the bound\nthat we got on d theta.", "start": 2825.56, "duration": 3.67}, {"text": "So this is the reason for\nchoosing that weird number", "start": 2829.23, "duration": 2.18}, {"text": "up there.", "start": 2831.41, "duration": 2.55}, {"text": "We find that the variation\nwithin each progression", "start": 2833.96, "duration": 16.09}, {"text": "is at most eta, right?", "start": 2850.05, "duration": 1.853}, {"text": "So the variation\nof this character", "start": 2851.903, "duration": 1.417}, {"text": "within each progression\nis not very large, OK?", "start": 2853.32, "duration": 2.855}, {"text": "And that's the claim.", "start": 2856.175, "duration": 0.875}, {"text": "Any questions?", "start": 2859.83, "duration": 1.95}, {"text": "All right, so this is the\nanalogous claim to the one", "start": 2861.78, "duration": 4.14}, {"text": "that we had--", "start": 2865.92, "duration": 0.86}, {"text": "the one that we used\nlast time, where", "start": 2866.78, "duration": 1.72}, {"text": "we said that the character\nis constant on each coset", "start": 2868.5, "duration": 3.45}, {"text": "of the hyperplane.", "start": 2871.95, "duration": 0.965}, {"text": "They're not exactly constant,\nbut almost good enough.", "start": 2872.915, "duration": 2.515}, {"text": "All right.", "start": 2885.61, "duration": 0.69}, {"text": "So the goal of step 2\nis to show an energy--", "start": 2886.3, "duration": 3.72}, {"text": "show a density increment, that\nif you have a large Fourier", "start": 2890.02, "duration": 3.0}, {"text": "coefficient, then\nwe want to claim", "start": 2893.02, "duration": 2.16}, {"text": "that the density\ngoes up significantly", "start": 2895.18, "duration": 3.54}, {"text": "on some subprogression.", "start": 2898.72, "duration": 3.22}, {"text": "And the next part,\nthe next lemma,", "start": 2901.94, "duration": 2.97}, {"text": "will get us to that goal.", "start": 2904.91, "duration": 2.37}, {"text": "And this part is very\nsimilar to the one", "start": 2907.28, "duration": 2.64}, {"text": "that we saw from last time,\nbut with this new partition", "start": 2909.92, "duration": 4.68}, {"text": "in mind, like I said.", "start": 2914.6, "duration": 1.38}, {"text": "If you have A that is\n3AP-free with density alpha,", "start": 2915.98, "duration": 11.88}, {"text": "and N is large\nenough, then there", "start": 2927.86, "duration": 10.13}, {"text": "exists some subprogression\nP. So by subprogression,", "start": 2937.99, "duration": 10.99}, {"text": "I just mean that I'm starting\nwith original progression", "start": 2948.98, "duration": 2.58}, {"text": "1 through N, and I'm zooming\ninto some subprogression,", "start": 2951.56, "duration": 3.78}, {"text": "with the length of P fairly\nlong, so the length of P", "start": 2955.34, "duration": 6.82}, {"text": "is at least cube root\nof N, and such that A,", "start": 2962.16, "duration": 7.27}, {"text": "when restricted to\nthis subprogression,", "start": 2969.43, "duration": 3.33}, {"text": "has a density increment.", "start": 2972.76, "duration": 3.535}, {"text": "OK, so originally, the\ndensity of A is alpha,", "start": 2981.995, "duration": 2.955}, {"text": "so we're zooming into\nsome subprogression P,", "start": 2984.95, "duration": 2.35}, {"text": "which is a pretty\nlong subprogression,", "start": 2987.3, "duration": 1.68}, {"text": "where the density\ngoes up significantly", "start": 2988.98, "duration": 1.89}, {"text": "from A to essentially A--", "start": 2990.87, "duration": 2.49}, {"text": "from alpha to roughly\nalpha plus alpha squared.", "start": 2993.36, "duration": 2.97}, {"text": "OK.", "start": 3000.24, "duration": 0.5}, {"text": "So we start with\nA, a 3AP-free set.", "start": 3007.12, "duration": 3.54}, {"text": "So from step 1, there exists\nsome theta with large--", "start": 3010.66, "duration": 14.45}, {"text": "so that corresponds to a\nlarge Fourier coefficient.", "start": 3025.11, "duration": 2.822}, {"text": "So this sum here is large.", "start": 3031.79, "duration": 14.28}, {"text": "OK, and now we use--", "start": 3048.782, "duration": 5.928}, {"text": "OK, so-- so step 1 obtains us,\nyou know, this consequence.", "start": 3054.71, "duration": 5.8}, {"text": "And from this theta, now we\napply the lemma up there to--", "start": 3060.51, "duration": 14.27}, {"text": "so we apply lemma\nwith, let's say,", "start": 3074.78, "duration": 1.98}, {"text": "eta being alpha squared over 30.", "start": 3076.76, "duration": 4.59}, {"text": "OK, so the exact constants\nare not so important.", "start": 3081.35, "duration": 2.34}, {"text": "But when we apply the\nlemma to partition,", "start": 3083.69, "duration": 7.32}, {"text": "and into a bunch\nof subprogressions,", "start": 3091.01, "duration": 6.71}, {"text": "which we'll call P1 through Pk.", "start": 3097.72, "duration": 2.85}, {"text": "And each of these\nprogressions have", "start": 3103.462, "duration": 4.618}, {"text": "length between cube root of\nN and twice cube root of N.", "start": 3108.08, "duration": 11.67}, {"text": "And I want to understand what\nhappens to the density of A", "start": 3119.75, "duration": 4.32}, {"text": "when restricted to\nthese progressions.", "start": 3124.07, "duration": 3.61}, {"text": "So starting with this\ninequality over here,", "start": 3127.68, "duration": 6.35}, {"text": "which suggests to us that\nthere must be some deviation.", "start": 3134.03, "duration": 3.12}, {"text": "OK, so starting\nwith what we saw.", "start": 3150.191, "duration": 4.509}, {"text": "And now, inside each\nprogression this e x theta", "start": 3154.7, "duration": 6.25}, {"text": "is roughly constant.", "start": 3160.95, "duration": 3.1}, {"text": "So if you pretend them\nas actually constant,", "start": 3164.05, "duration": 2.71}, {"text": "I can break up the sum,\ndepending on where the x's lie.", "start": 3166.76, "duration": 7.47}, {"text": "So i from 1 to k.", "start": 3174.23, "duration": 3.3}, {"text": "And let me sum inside\neach progression.", "start": 3177.53, "duration": 3.962}, {"text": "So by triangle inequality, I\ncan upper bound the first sum", "start": 3193.56, "duration": 4.05}, {"text": "by where I now cut the sum into\nprogression by progression.", "start": 3197.61, "duration": 5.93}, {"text": "And on each progression, this\ncharacter is roughly constant.", "start": 3203.54, "duration": 5.98}, {"text": "So let me take out the\nmaximum possible deviations", "start": 3209.52, "duration": 5.24}, {"text": "from them being constant.", "start": 3214.76, "duration": 1.68}, {"text": "So upper bound-- again, you'll\nfind that we can essentially", "start": 3216.44, "duration": 12.085}, {"text": "pretend--", "start": 3228.525, "duration": 0.5}, {"text": "all right, so if\neach exponential", "start": 3232.19, "duration": 3.22}, {"text": "is constant on each\nsubprogression,", "start": 3235.41, "duration": 2.04}, {"text": "then I might as well\njust have this sum here.", "start": 3237.45, "duration": 3.3}, {"text": "But I lose a little bit, because\nit's not exactly constant.", "start": 3240.75, "duration": 2.76}, {"text": "It's almost constant.", "start": 3243.51, "duration": 1.17}, {"text": "So I loose a little bit.", "start": 3244.68, "duration": 1.46}, {"text": "And that little bit is this eta.", "start": 3246.14, "duration": 5.21}, {"text": "So you lose that\nlittle bit of eta.", "start": 3251.35, "duration": 2.09}, {"text": "And so on each\nprogression, P i, you", "start": 3253.44, "duration": 5.52}, {"text": "lose at most something that's\nessentially of alpha squared", "start": 3258.96, "duration": 3.87}, {"text": "times the length of P i.", "start": 3262.83, "duration": 1.435}, {"text": "OK.", "start": 3267.24, "duration": 1.92}, {"text": "Now, you see, I've chosen\nthe error parameter", "start": 3269.16, "duration": 3.72}, {"text": "so that everything\nI've lost is not", "start": 3272.88, "duration": 4.38}, {"text": "so much more than the\ninitial bound I began with.", "start": 3277.26, "duration": 3.99}, {"text": "So in particular,\nwe see that even", "start": 3281.25, "duration": 5.85}, {"text": "if we had pretended that the\ncharacters were constant,", "start": 3287.1, "duration": 2.7}, {"text": "on each progression we would\nhave still obtained some lower", "start": 3289.8, "duration": 4.29}, {"text": "bound of the total deviation.", "start": 3294.09, "duration": 2.79}, {"text": "OK.", "start": 3308.68, "duration": 1.18}, {"text": "And what is this\nquantity over here?", "start": 3309.86, "duration": 5.27}, {"text": "Oh, you see, I'm\nrestricting each sum", "start": 3315.13, "duration": 3.75}, {"text": "to each subprogression,\nbut the sum", "start": 3318.88, "duration": 4.71}, {"text": "here, even though it's\nthe sum, but it's really", "start": 3323.59, "duration": 2.85}, {"text": "counting how many elements\nof A are in that progression.", "start": 3326.44, "duration": 5.55}, {"text": "So this sum over here\nis the same thing.", "start": 3331.99, "duration": 3.72}, {"text": "OK, so let me write\nit in a new board.", "start": 3335.71, "duration": 2.833}, {"text": "Oh, we don't need\nstep 1 anymore.", "start": 3343.48, "duration": 1.733}, {"text": "All right.", "start": 3349.077, "duration": 2.193}, {"text": "So what we have--", "start": 3351.27, "duration": 1.17}, {"text": "OK, so left-hand side over\nthere is this quantity here,", "start": 3361.386, "duration": 4.045}, {"text": "all right?", "start": 3365.431, "duration": 2.259}, {"text": "We see that the right-hand side,\neven though you have that sum,", "start": 3367.69, "duration": 8.2}, {"text": "it is really just counting\nhow many elements of A", "start": 3375.89, "duration": 3.12}, {"text": "are in each progression versus\nhow many you should expect", "start": 3379.01, "duration": 4.35}, {"text": "based on the overall\ndensity of A. OK,", "start": 3383.36, "duration": 5.39}, {"text": "so that should look similar\nto what we got last time.", "start": 3388.75, "duration": 3.47}, {"text": "I know the intuition\nshould be that, well,", "start": 3392.22, "duration": 3.06}, {"text": "if the average deviation\nis large, then one of them,", "start": 3395.28, "duration": 6.93}, {"text": "one of these terms, should\nhave the density increment.", "start": 3402.21, "duration": 5.237}, {"text": "If you try to do the next\nstep somewhat naively,", "start": 3411.19, "duration": 2.76}, {"text": "you run into an issue,\nbecause it could be--", "start": 3413.95, "duration": 2.89}, {"text": "now, here you have k terms.", "start": 3416.84, "duration": 2.45}, {"text": "It could be that you have all\nthe densities except for one", "start": 3419.29, "duration": 7.98}, {"text": "going up only slightly, and one\ndensity dropping dramatically,", "start": 3427.27, "duration": 5.37}, {"text": "in which case you may not\nhave a significant density", "start": 3432.64, "duration": 2.94}, {"text": "increment, all right?", "start": 3435.58, "duration": 2.63}, {"text": "So we want to show that\non some progression", "start": 3438.21, "duration": 2.63}, {"text": "the density increases\nsignificantly.", "start": 3440.84, "duration": 3.51}, {"text": "So far, from this\ninequality, we just", "start": 3444.35, "duration": 2.52}, {"text": "know that there is some\nsubprogression where", "start": 3446.87, "duration": 2.85}, {"text": "the density changes\nsignificantly.", "start": 3449.72, "duration": 2.73}, {"text": "But of course, the overall\ndensity, the average density,", "start": 3452.45, "duration": 2.4}, {"text": "should remain constant.", "start": 3454.85, "duration": 1.27}, {"text": "So if some goes up,\nothers must go down.", "start": 3456.12, "duration": 2.83}, {"text": "But if you just try to\ndo an averaging argument,", "start": 3458.95, "duration": 2.23}, {"text": "you have to be careful, OK?", "start": 3461.18, "duration": 1.71}, {"text": "So there was a trick last\ntime, which we didn't really", "start": 3462.89, "duration": 2.4}, {"text": "need last time, but now,\nit's much more useful, where", "start": 3465.29, "duration": 4.62}, {"text": "I want to show\nthat if this holds,", "start": 3469.91, "duration": 2.31}, {"text": "then some P i sees\na large energy--", "start": 3472.22, "duration": 3.63}, {"text": "sees a large density increment.", "start": 3475.85, "duration": 2.67}, {"text": "And to do that, let me rewrite\nthe sum as the following,", "start": 3478.52, "duration": 10.47}, {"text": "so I keep the same expression.", "start": 3488.99, "duration": 2.67}, {"text": "And I add a term, which\nis the same thing,", "start": 3494.88, "duration": 4.1}, {"text": "but without the absolute value.", "start": 3498.98, "duration": 1.58}, {"text": "OK, so you see these\nguys, they total to 0,", "start": 3509.434, "duration": 5.566}, {"text": "so adding that term doesn't\nchange my expression.", "start": 3515.0, "duration": 5.35}, {"text": "But now, the summand\nis always non-negative.", "start": 3520.35, "duration": 5.11}, {"text": "So it's either 0 or\ntwice this number,", "start": 3525.46, "duration": 3.92}, {"text": "depending on the\nsign of that number.", "start": 3529.38, "duration": 2.12}, {"text": "OK.", "start": 3534.61, "duration": 0.5}, {"text": "So comparing left-hand\nside and right-hand side,", "start": 3535.11, "duration": 2.49}, {"text": "we see that there\nmust be some i.", "start": 3537.6, "duration": 3.19}, {"text": "So hence, there exists some eye\nsuch that the left-hand side--", "start": 3540.79, "duration": 7.94}, {"text": "the i-th term on\nthe left hand side", "start": 3553.41, "duration": 1.86}, {"text": "is less than or equal\nto the i-th term", "start": 3555.27, "duration": 3.002}, {"text": "on the right-hand side.", "start": 3558.272, "duration": 0.958}, {"text": "And in particular, that\nterm should be positive,", "start": 3564.94, "duration": 3.7}, {"text": "so it implies--", "start": 3568.64, "duration": 2.82}, {"text": "OK, so how can you\nget this inequality?", "start": 3571.46, "duration": 4.28}, {"text": "It implies simply that the\nrestriction of a to this P i", "start": 3575.74, "duration": 4.8}, {"text": "is at least alpha plus alpha\nsquared over 40 times P i.", "start": 3580.54, "duration": 10.27}, {"text": "So this claim here just says\nthat on the i-th progression,", "start": 3590.81, "duration": 3.51}, {"text": "there's a significant\nenergy increment.", "start": 3594.32, "duration": 4.25}, {"text": "If it's more decrement,\nthat term would have been 0.", "start": 3598.57, "duration": 3.772}, {"text": "So remember that.", "start": 3602.342, "duration": 2.268}, {"text": "OK.", "start": 3616.04, "duration": 1.24}, {"text": "So this achieves what we\nwere looking for in step 2,", "start": 3617.28, "duration": 3.33}, {"text": "namely to find that there\nis a density increment", "start": 3620.61, "duration": 4.11}, {"text": "on some long subprogression.", "start": 3624.72, "duration": 3.321}, {"text": "OK, so now we can go to\nstep 3, which is basically", "start": 3628.041, "duration": 2.589}, {"text": "the same as what we saw\nlast time, where now we", "start": 3630.63, "duration": 4.83}, {"text": "want to iterate this\ndensity increment.", "start": 3635.46, "duration": 3.469}, {"text": "OK, so it's basically the\nsame argument as last time,", "start": 3647.911, "duration": 7.989}, {"text": "but you start with\ndensity alpha,", "start": 3655.9, "duration": 4.4}, {"text": "and each step in the\niteration, the density", "start": 3660.3, "duration": 3.84}, {"text": "goes up quite a bit.", "start": 3664.14, "duration": 1.46}, {"text": "And we want to control\nthe total number of steps,", "start": 3669.74, "duration": 4.29}, {"text": "knowing that the final\ndensity is at most 1, always.", "start": 3674.03, "duration": 6.674}, {"text": "OK.", "start": 3680.704, "duration": 2.666}, {"text": "So how many steps can you take?", "start": 3683.37, "duration": 1.65}, {"text": "Right, so this was the same\nargument that we saw last time.", "start": 3685.02, "duration": 3.06}, {"text": "We see that starting with\nalpha, alpha 0 being alpha,", "start": 3688.08, "duration": 10.37}, {"text": "it doubles after a certain\nnumber of steps, right?", "start": 3698.45, "duration": 3.36}, {"text": "So we double after--", "start": 3701.81, "duration": 3.78}, {"text": "OK, so how many\nsteps do you need?", "start": 3705.59, "duration": 3.12}, {"text": "Well, I want to get\nfrom alpha to 2 alpha.", "start": 3708.71, "duration": 4.72}, {"text": "So I need at most\nalpha over 40 steps.", "start": 3713.43, "duration": 7.53}, {"text": "OK, so last time I\nwas slightly sloppy.", "start": 3720.96, "duration": 2.81}, {"text": "And so there's basically a\nfloor, upper floor down--", "start": 3723.77, "duration": 5.86}, {"text": "rounding up or down situation.", "start": 3729.63, "duration": 1.43}, {"text": "But I should add a plus 1.", "start": 3731.06, "duration": 2.56}, {"text": "Yeah.", "start": 3733.62, "duration": 0.5}, {"text": "AUDIENCE: Shouldn't\nit be 40 over alpha?", "start": 3734.12, "duration": 2.0}, {"text": "YUFEI ZHAO: 40-- thank you.", "start": 3736.12, "duration": 1.22}, {"text": "40 over alpha, yeah.", "start": 3737.34, "duration": 2.59}, {"text": "So you double after at\nmost that many steps.", "start": 3739.93, "duration": 5.15}, {"text": "And then now, you add\ndensity at least 2 alpha.", "start": 3745.08, "duration": 6.48}, {"text": "So we double after at most 20\nover alpha steps, and so on.", "start": 3751.56, "duration": 8.32}, {"text": "And we double at most--", "start": 3759.88, "duration": 4.81}, {"text": "well, basically, log sub\n2 of 1 over alpha times.", "start": 3764.69, "duration": 7.0}, {"text": "OK, so anyway, putting\neverything together,", "start": 3775.658, "duration": 2.942}, {"text": "we see that the\ntotal number of steps", "start": 3778.6, "duration": 5.75}, {"text": "is at most on the\norder of 1 over alpha.", "start": 3784.35, "duration": 4.44}, {"text": "When you stop, you can\nonly stop for one reason,", "start": 3792.34, "duration": 3.15}, {"text": "because in the--", "start": 3795.49, "duration": 2.72}, {"text": "yeah.", "start": 3798.21, "duration": 0.5}, {"text": "So, yeah.", "start": 3798.71, "duration": 1.17}, {"text": "So in step 1,\nremember, the iteration", "start": 3799.88, "duration": 4.91}, {"text": "said that the\nprocess terminates.", "start": 3804.79, "duration": 3.972}, {"text": "The process can always go\non, and then terminates", "start": 3808.762, "duration": 8.827}, {"text": "if the length--", "start": 3817.589, "duration": 5.581}, {"text": "so you're now at\nstep i, so let N i be", "start": 3823.17, "duration": 2.42}, {"text": "the length of the progression,\nthat step i is at most C times", "start": 3825.59, "duration": 8.99}, {"text": "alpha i to the minus 12th.", "start": 3834.58, "duration": 2.41}, {"text": "So we have a--", "start": 3836.99, "duration": 4.19}, {"text": "right, so provided\nthat N is large enough,", "start": 3841.18, "duration": 3.8}, {"text": "you can always pass\nto a subprogression.", "start": 3844.98, "duration": 2.04}, {"text": "And here, when you pass to\nsubprogression, of course,", "start": 3847.02, "duration": 2.52}, {"text": "you can re-label\nthat subprogression.", "start": 3849.54, "duration": 2.49}, {"text": "And it's now, you\nknow, 1 through N i.", "start": 3852.03, "duration": 3.11}, {"text": "Right, so I can-- it's--", "start": 3855.14, "duration": 1.413}, {"text": "all the progressions\nare basically", "start": 3856.553, "duration": 1.417}, {"text": "the same as the first\nset of positive integers.", "start": 3857.97, "duration": 3.403}, {"text": "I'm sorry, prefix of\nthe positive integers.", "start": 3861.373, "duration": 2.477}, {"text": "So when we stop a\nstep i, you must", "start": 3863.85, "duration": 2.74}, {"text": "have N sub i being at most\nthis quantity over here, which", "start": 3866.59, "duration": 4.7}, {"text": "is at most C times the initial\ndensity raised to this minus", "start": 3871.29, "duration": 5.22}, {"text": "12th.", "start": 3876.51, "duration": 0.94}, {"text": "So therefore, the initial length\nN of the space is bounded by--", "start": 3877.45, "duration": 11.6}, {"text": "well, each time we went\ndown by a cube root at most.", "start": 3889.05, "duration": 6.545}, {"text": "Right, so the fine--", "start": 3895.595, "duration": 1.965}, {"text": "if you stop a step i,\nthen the initial length", "start": 3897.56, "duration": 3.42}, {"text": "is at most N sub i to the\n3 times 3 to the power of i", "start": 3900.98, "duration": 6.69}, {"text": "each time you're\ndoing a cube root.", "start": 3907.67, "duration": 3.252}, {"text": "OK, so you put\neverything together.", "start": 3910.922, "duration": 2.213}, {"text": "At most that many\niterations when you stop", "start": 3920.28, "duration": 2.08}, {"text": "the length is at most this.", "start": 3922.36, "duration": 2.421}, {"text": "So you put them\ntogether, and then you", "start": 3924.781, "duration": 1.899}, {"text": "find that the N must be at\nmost double exponential in 1", "start": 3926.68, "duration": 10.64}, {"text": "over the density.", "start": 3937.32, "duration": 2.82}, {"text": "In other words, the\ndensity is at most 1", "start": 3940.14, "duration": 7.53}, {"text": "over log log N, which is what\nwe claimed in Roth's theorem,", "start": 3947.67, "duration": 7.995}, {"text": "so what we claimed up there.", "start": 3955.665, "duration": 1.395}, {"text": "OK.", "start": 3961.46, "duration": 0.5}, {"text": "So that finishes the proof.", "start": 3961.96, "duration": 1.268}, {"text": "Any questions?", "start": 3972.21, "duration": 1.097}, {"text": "So the message here is that it's\nthe same proof as last time,", "start": 3978.78, "duration": 3.45}, {"text": "but we need to do\na bit more work.", "start": 3982.23, "duration": 2.43}, {"text": "And none of this\nwork is difficult,", "start": 3984.66, "duration": 1.56}, {"text": "but there are more technical.", "start": 3986.22, "duration": 1.56}, {"text": "And that's often\nthe theme that you", "start": 3987.78, "duration": 1.8}, {"text": "see in additive combinatorics.", "start": 3989.58, "duration": 1.92}, {"text": "This is part of the reason\nwhy the finite field model is", "start": 3991.5, "duration": 3.15}, {"text": "a really nice playground,\nbecause there, things", "start": 3994.65, "duration": 3.48}, {"text": "tend to be often cleaner,\nbut the idea's often similar,", "start": 3998.13, "duration": 3.72}, {"text": "or the same ideas.", "start": 4001.85, "duration": 1.29}, {"text": "Not always.", "start": 4003.14, "duration": 0.75}, {"text": "Next lecture, we'll\nsee one technique", "start": 4003.89, "duration": 2.352}, {"text": "where there's a\ndramatic difference", "start": 4006.242, "duration": 1.458}, {"text": "between the finite field vector\nspace and over the integers.", "start": 4007.7, "duration": 3.24}, {"text": "But for many things in\nadditive combinatorics,", "start": 4010.94, "duration": 2.23}, {"text": "the finite field\nvector space is just", "start": 4013.17, "duration": 1.55}, {"text": "a nicer place to be in to try\nall your ideas and techniques.", "start": 4014.72, "duration": 3.452}, {"text": "Let me comment on some analogies\nbetween these two approaches", "start": 4024.44, "duration": 4.65}, {"text": "and compare the bounds.", "start": 4029.09, "duration": 1.11}, {"text": "So on one hand--", "start": 4033.09, "duration": 2.19}, {"text": "OK, so we saw last time\nthis proof in F3 to the N,", "start": 4035.28, "duration": 4.85}, {"text": "and now in the integers\ninside this interval of length", "start": 4040.13, "duration": 6.64}, {"text": "N. So let me write\nuppercase N in both cases", "start": 4046.77, "duration": 6.51}, {"text": "to be the size of the\noverall ambient space.", "start": 4053.28, "duration": 4.925}, {"text": "OK, so what kind of bounds\ndo we get in both situations?", "start": 4058.205, "duration": 5.105}, {"text": "So last time, for--", "start": 4063.31, "duration": 2.74}, {"text": "in F3 to the N, we\ngot a bound which", "start": 4066.05, "duration": 5.67}, {"text": "is of the order N over log N,\nwhereas today, the bound is", "start": 4071.72, "duration": 14.5}, {"text": "somewhat worse.", "start": 4086.22, "duration": 2.34}, {"text": "It's a little bit worse.", "start": 4088.56, "duration": 1.36}, {"text": "Now we lose an extra log.", "start": 4089.92, "duration": 3.77}, {"text": "So where do we lose an\nextra log in this argument?", "start": 4093.69, "duration": 3.12}, {"text": "So where does these\ntwo argument--", "start": 4096.81, "duration": 2.25}, {"text": "where do these two arguments\ndiffer, quantitatively?", "start": 4099.06, "duration": 3.92}, {"text": "Yep?", "start": 4106.24, "duration": 0.5}, {"text": "AUDIENCE: When you're dividing\nby 3 versus [INAUDIBLE]??", "start": 4106.74, "duration": 2.829}, {"text": "YUFEI ZHAO: OK, so\nyou're dividing by--", "start": 4109.569, "duration": 1.991}, {"text": "so here, in each\niteration, over here,", "start": 4111.56, "duration": 10.4}, {"text": "your size of the iteration--", "start": 4121.96, "duration": 4.439}, {"text": "I mean, each iteration\nthe size of the space", "start": 4126.399, "duration": 1.901}, {"text": "goes down by a factor of\n3, whereas over here, it", "start": 4128.3, "duration": 4.979}, {"text": "could go down by a cube root.", "start": 4133.279, "duration": 3.7}, {"text": "And that's precisely right.", "start": 4136.979, "duration": 1.42}, {"text": "So that explains for this\nextra log in the balance.", "start": 4138.399, "duration": 4.021}, {"text": "So while this is\na great analogy,", "start": 4145.63, "duration": 2.86}, {"text": "it's not a perfect analogy.", "start": 4148.49, "duration": 2.27}, {"text": "You see there is this\ndivergence here between the two", "start": 4150.76, "duration": 2.519}, {"text": "situations.", "start": 4153.279, "duration": 1.511}, {"text": "And so then you\nmight ask, is there", "start": 4154.79, "duration": 1.64}, {"text": "some way to avoid the\nloss, this extra log factor", "start": 4156.43, "duration": 4.81}, {"text": "loss over here?", "start": 4161.24, "duration": 1.71}, {"text": "Is there some way to\ncarry out the strategy", "start": 4162.95, "duration": 2.31}, {"text": "that we did last\ntime in a way that", "start": 4165.26, "duration": 3.429}, {"text": "is much more faithful to\nthat strategy of passing down", "start": 4168.689, "duration": 2.611}, {"text": "to subspaces.", "start": 4171.3, "duration": 1.76}, {"text": "So here, we pass\nto progressions.", "start": 4173.06, "duration": 3.03}, {"text": "And because we have to do this\nextra pigeonhole-type argument,", "start": 4176.09, "duration": 4.559}, {"text": "it was somewhat lost--", "start": 4180.649, "duration": 1.321}, {"text": "we lost a power, which\ntranslated into this extra log.", "start": 4181.97, "duration": 6.8}, {"text": "So it turns out there\nis some way to do this.", "start": 4188.77, "duration": 3.14}, {"text": "So let me just\nbriefly mention what's", "start": 4191.91, "duration": 1.69}, {"text": "the idea that is\ninvolved, all right?", "start": 4193.6, "duration": 2.04}, {"text": "So last time, we\nwent down from--", "start": 4195.64, "duration": 4.29}, {"text": "so the main objects\nthat we were passing", "start": 4199.93, "duration": 2.73}, {"text": "would start with a vector\nspace and pass down", "start": 4202.66, "duration": 2.52}, {"text": "to a subspace, which is\nalso a vector space, right?", "start": 4205.18, "duration": 3.13}, {"text": "So you can define subspaces in\nF3 to the N by the following.", "start": 4208.31, "duration": 9.18}, {"text": "So I can start with some set\nof characters U, and I define--", "start": 4217.49, "duration": 6.82}, {"text": "some set of characters\nS, and I define", "start": 4224.31, "duration": 2.32}, {"text": "U sub S to be basically the\northogonal complement of S.", "start": 4226.63, "duration": 14.192}, {"text": "OK, so this is a subspace.", "start": 4240.822, "duration": 2.15}, {"text": "And these were the\nkind of subspace", "start": 4242.972, "duration": 1.458}, {"text": "that we saw last time, because\nthe S's or the R's that", "start": 4244.43, "duration": 3.738}, {"text": "came out of the proof last\ntime, every time we saw one,", "start": 4248.168, "duration": 2.292}, {"text": "we threw it in.", "start": 4250.46, "duration": 0.93}, {"text": "We cut down to a smaller\nsubspace, and we repeat.", "start": 4251.39, "duration": 4.94}, {"text": "But the progressions, they\ndon't really look like this.", "start": 4256.33, "duration": 2.688}, {"text": "So the question is,\nis there some way", "start": 4259.018, "duration": 1.542}, {"text": "to do this argument so that\nyou end up with progressions,", "start": 4260.56, "duration": 2.375}, {"text": "and looked like that?", "start": 4262.935, "duration": 1.075}, {"text": "And it turns out there is a way.", "start": 4264.01, "duration": 2.29}, {"text": "And there are these\nobjects, which", "start": 4266.3, "duration": 1.85}, {"text": "we'll see more later in this\ncourse, called Bohr sets.", "start": 4268.15, "duration": 5.065}, {"text": "OK, so they were\nused by Bourgain", "start": 4273.215, "duration": 4.815}, {"text": "to mimic this Machoulin argument\nthat we saw last time more", "start": 4278.03, "duration": 5.19}, {"text": "faithfully into\nthe integers, where", "start": 4283.22, "duration": 2.79}, {"text": "we're going to come up with some\nset of integers that resemble--", "start": 4286.01, "duration": 4.83}, {"text": "much more closely resemble\nthis notion of subspaces", "start": 4290.84, "duration": 4.11}, {"text": "in the finite field setting.", "start": 4294.95, "duration": 2.34}, {"text": "And for this, it's much\neasier to work inside a group.", "start": 4297.29, "duration": 3.73}, {"text": "So instead of working\nin the integers,", "start": 4301.02, "duration": 1.88}, {"text": "let's work inside and Z mod nZ.", "start": 4302.9, "duration": 2.21}, {"text": "So we can do Fourier\ntransform in Z mod nZ, so", "start": 4305.11, "duration": 2.47}, {"text": "the discrete Fourier\nanalysis here.", "start": 4307.58, "duration": 2.46}, {"text": "So in Z mod nZ, we define--", "start": 4310.04, "duration": 2.295}, {"text": "so given a S, let's\ndefine this Bohr", "start": 4315.51, "duration": 3.75}, {"text": "set to be the set of\nelements of Z mod nZ such", "start": 4319.26, "duration": 14.82}, {"text": "that if you look at what\nreally is supposed to resemble", "start": 4334.08, "duration": 6.39}, {"text": "this thing over here, OK?", "start": 4340.47, "duration": 5.81}, {"text": "If this quantity is\nsmall for all S--", "start": 4346.28, "duration": 8.0}, {"text": "OK, so we put that element\ninto this Bohr set.", "start": 4354.28, "duration": 2.65}, {"text": "OK, so these sets, they function\nmuch more like subspaces.", "start": 4359.67, "duration": 5.2}, {"text": "So there are the\nanalog of subspaces", "start": 4364.87, "duration": 4.02}, {"text": "inside Z mod nZ, which,\nyou know, n is prime,", "start": 4368.89, "duration": 4.53}, {"text": "has no subgroups.", "start": 4373.42, "duration": 1.01}, {"text": "It has no natural\nsubspace structure.", "start": 4374.43, "duration": 2.56}, {"text": "But by looking at\nthese Bohr sets,", "start": 4376.99, "duration": 1.89}, {"text": "they provide a\nnatural way to set up", "start": 4378.88, "duration": 4.14}, {"text": "this argument so that you can--", "start": 4383.02, "duration": 2.31}, {"text": "but with much more\ntechnicalities,", "start": 4385.33, "duration": 2.58}, {"text": "repeat these kind of arguments\nmore similar to last time,", "start": 4387.91, "duration": 5.72}, {"text": "but passing not to\nsubspaces but to Bohr sets.", "start": 4393.63, "duration": 3.59}, {"text": "And then with quite\na bit of extra work,", "start": 4397.22, "duration": 5.05}, {"text": "one can obtain bounds of the\nquantity N over a poly log N.", "start": 4402.27, "duration": 10.1}, {"text": "So the current best bound\nI mentioned last time", "start": 4412.37, "duration": 4.14}, {"text": "is of this type, which is\nthrough further refinements", "start": 4416.51, "duration": 3.57}, {"text": "of this technique.", "start": 4420.08, "duration": 1.267}, {"text": "The last thing I\nwant to mention today", "start": 4430.61, "duration": 1.62}, {"text": "is, so far we've been\ntalking about 3APs.", "start": 4432.23, "duration": 3.81}, {"text": "So what about four term\narithmetic progressions?", "start": 4436.04, "duration": 3.56}, {"text": "OK, do any of the things that we\ntalk about here work for 4APs?", "start": 4439.6, "duration": 5.398}, {"text": "And there's an analogy to be\nmade here compared to what", "start": 4444.998, "duration": 2.292}, {"text": "we discussed with graphs.", "start": 4447.29, "duration": 1.795}, {"text": "So in graphs, we had a triangle\ncounting lemma and a triangle", "start": 4449.085, "duration": 2.685}, {"text": "removal lemma.", "start": 4451.77, "duration": 1.16}, {"text": "And then we said\nthat to prove 4APs,", "start": 4452.93, "duration": 2.28}, {"text": "we would need the hypergraph\nversion, the simplex removal", "start": 4455.21, "duration": 3.18}, {"text": "lemma, hypergraph\nregularity lemma.", "start": 4458.39, "duration": 2.19}, {"text": "And that was much\nmore difficult.", "start": 4460.58, "duration": 1.92}, {"text": "And that analogy\ncarries through,", "start": 4462.5, "duration": 1.67}, {"text": "and the same kind of\ndifficulties that come up.", "start": 4464.17, "duration": 2.87}, {"text": "So it can be done, but\nyou need something more.", "start": 4467.04, "duration": 3.66}, {"text": "And the main message I\nwant you to take away", "start": 4470.7, "duration": 3.89}, {"text": "is that 4APs, while we\nhad a counting lemma that", "start": 4474.59, "duration": 8.35}, {"text": "says that the\nFourier coefficients,", "start": 4482.94, "duration": 3.04}, {"text": "so the Fourier transform,\ncontrols 3AP counts,", "start": 4485.98, "duration": 8.56}, {"text": "it turns out the same\nis not true for 4APs.", "start": 4494.54, "duration": 4.56}, {"text": "So the Fourier does\nnot control 4AP counts.", "start": 4499.1, "duration": 6.1}, {"text": "Let me give you some--", "start": 4510.41, "duration": 1.93}, {"text": "OK.", "start": 4512.34, "duration": 0.5}, {"text": "So in fact, in the\nhomework for this week,", "start": 4512.84, "duration": 2.07}, {"text": "there's a specific example of\na set where it has uniformly", "start": 4514.91, "duration": 4.96}, {"text": "small Fourier coefficients.", "start": 4519.87, "duration": 1.62}, {"text": "But that's the wrong\nnumber of 4APs.", "start": 4521.49, "duration": 1.5}, {"text": "So the following-- it is true--", "start": 4527.13, "duration": 1.37}, {"text": "OK, so it is true that you\nhave Szemer\u00e9di's term in--", "start": 4528.5, "duration": 4.11}, {"text": "let's just talk about\nthe finite field setting,", "start": 4532.61, "duration": 2.16}, {"text": "where things are a\nbit easier to discuss.", "start": 4534.77, "duration": 2.07}, {"text": "So it is true that the size\nof the biggest subset of F5", "start": 4536.84, "duration": 5.67}, {"text": "to the N is a tiny\nfraction-- it's a little, one", "start": 4542.51, "duration": 2.67}, {"text": "fraction of the entire space.", "start": 4545.18, "duration": 1.35}, {"text": "OK, I use F5 here,\nbecause if I set F3,", "start": 4546.53, "duration": 3.93}, {"text": "it doesn't make sense\nto talk about 4APs.", "start": 4550.46, "duration": 3.23}, {"text": "So F5, but it doesn't really\nmatter which specific field.", "start": 4553.69, "duration": 5.3}, {"text": "So you can prove this using\nhypergraph removal, same proof,", "start": 4558.99, "duration": 4.56}, {"text": "verbatim, that we saw earlier,\nif you have hypergraph removal.", "start": 4563.55, "duration": 3.78}, {"text": "But if you want to try to prove\nit using Fourier analysis,", "start": 4567.33, "duration": 3.28}, {"text": "well, it doesn't work quite\nusing the same strategy.", "start": 4570.61, "duration": 5.03}, {"text": "But in fact, there\nis a modification", "start": 4575.64, "duration": 1.95}, {"text": "that would allow\nyou to make it work.", "start": 4577.59, "duration": 3.21}, {"text": "But you need an extension\nof Fourier analysis.", "start": 4580.8, "duration": 2.87}, {"text": "And it is known as higher\norder Fourier analysis, which", "start": 4583.67, "duration": 7.12}, {"text": "was an important development in\nmodern additive combinatorics", "start": 4590.79, "duration": 4.68}, {"text": "that initially arose\nin Gowers' work", "start": 4595.47, "duration": 3.12}, {"text": "where he gave a new proof\nof similarities theorem.", "start": 4598.59, "duration": 2.6}, {"text": "So Gowers didn't\nwork in this setting.", "start": 4601.19, "duration": 1.69}, {"text": "He worked in integers.", "start": 4602.88, "duration": 1.14}, {"text": "But many of the ideas\noriginated from his paper,", "start": 4604.02, "duration": 3.12}, {"text": "and then subsequently\ndeveloped by a lot", "start": 4607.14, "duration": 1.98}, {"text": "of people in various settings.", "start": 4609.12, "duration": 2.13}, {"text": "I just want to give you one\nspecific statement, what", "start": 4611.25, "duration": 2.627}, {"text": "this high-order Fourier\nanalysis looks like.", "start": 4613.877, "duration": 1.833}, {"text": "So it's a fancy term,\nand the statements often", "start": 4615.71, "duration": 2.74}, {"text": "get very technical.", "start": 4618.45, "duration": 1.62}, {"text": "But I just want to give you one\nconcrete thing to take away.", "start": 4620.07, "duration": 4.07}, {"text": "All right, so for\na Fourier piece,", "start": 4624.14, "duration": 1.65}, {"text": "higher-order Fourier\nanalysis, roughly--", "start": 4625.79, "duration": 2.43}, {"text": "OK, so it also goes by the name\nquadratic Fourier analysis.", "start": 4628.22, "duration": 4.264}, {"text": "OK, so let me give you\na very specific instance", "start": 4636.436, "duration": 5.934}, {"text": "of the theorem.", "start": 4642.37, "duration": 1.2}, {"text": "And this can be sometimes\ncalled an inverse theorem", "start": 4647.09, "duration": 4.62}, {"text": "for quadratic Fourier analysis.", "start": 4651.71, "duration": 2.562}, {"text": "OK, so for every delta,\nthere exists some c such", "start": 4660.79, "duration": 7.91}, {"text": "that the following is true.", "start": 4668.7, "duration": 2.79}, {"text": "If A is a subset of a F5 to the\nN with density alpha and such", "start": 4671.49, "duration": 11.9}, {"text": "that it's--", "start": 4683.39, "duration": 1.742}, {"text": "OK, so now lambda\nsub 4, so this is", "start": 4685.132, "duration": 2.618}, {"text": "the 4AP density,\nso similar to 3AP,", "start": 4687.75, "duration": 2.63}, {"text": "but now you write four terms.", "start": 4690.38, "duration": 1.33}, {"text": "The 4AP density of A differs\nfrom alpha to the fourth", "start": 4691.71, "duration": 6.92}, {"text": "by a significant amount.", "start": 4698.63, "duration": 2.912}, {"text": "OK, so for 3APs, then we said\nthat now A has a large Fourier", "start": 4701.542, "duration": 3.508}, {"text": "coefficient, right?", "start": 4705.05, "duration": 1.7}, {"text": "So for-- OK.", "start": 4709.7, "duration": 3.94}, {"text": "For 4APs, that may not be\ntrue, but the following", "start": 4713.64, "duration": 3.89}, {"text": "is true, right?", "start": 4717.53, "duration": 0.845}, {"text": "So then there exists a\nnon-zero quadratic polynomial.", "start": 4718.375, "duration": 10.293}, {"text": "F and N variables over F5\nsuch that the indicator", "start": 4735.5, "duration": 13.12}, {"text": "function of A correlates with\nthis quadratic exponential", "start": 4748.62, "duration": 4.66}, {"text": "face.", "start": 4753.28, "duration": 0.5}, {"text": "So Fourier analysis,\nthe conclusion", "start": 4763.2, "duration": 1.633}, {"text": "that we got from\ncounting lemma is", "start": 4764.833, "duration": 1.417}, {"text": "that you have some\nlinear function", "start": 4766.25, "duration": 2.19}, {"text": "F, such that this\nquantity is large,", "start": 4768.44, "duration": 5.21}, {"text": "this large Fourier coefficient.", "start": 4773.65, "duration": 1.98}, {"text": "OK, so that is not true 4APs.", "start": 4775.63, "duration": 2.35}, {"text": "But what is true\nis that now you can", "start": 4777.98, "duration": 1.89}, {"text": "look at quadratic exponential\nfaces, and then it is true.", "start": 4779.87, "duration": 4.99}, {"text": "So that's the content\nof higher order Fourier.", "start": 4784.86, "duration": 3.86}, {"text": "I mean, that's the example of\nhigher-order Fourier analysis.", "start": 4788.72, "duration": 2.94}, {"text": "And you can imagine with\nthis type of result,", "start": 4791.66, "duration": 4.35}, {"text": "and with quite a\nbit more work, you", "start": 4796.01, "duration": 2.19}, {"text": "can try to follow a\nsimilar density increment", "start": 4798.2, "duration": 3.36}, {"text": "strategy to prove\nsimilarities term for 4APs.", "start": 4801.56, "duration": 5.55}]