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Arithmetic Combinatorics, Gowers Norms, and Nilspaces

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Arithmetic Combinatorics, Gowers Norms, and Nilspaces Arithmetic combinatorics, Gowers norms, and nilspaces Pablo Candela (UAM, ICMAT) JAE School 2019, ICMAT 1 / 47 Introduction A central theme in arithmetic combinatorics is the study of the occurrence of combinatorial structures in subsets of abelian groups, under assumptions on the subsets that are typically quite weak. A major result in this theme: Theorem (Szemer´edi,1975 [9]) For every positive integer k and every α > 0 there exists N0 > 0 such that the following holds. For every integer N > N0, every subset A ⊂ [N] of cardinality at least αN contains an arithmetic progression of length k. Here [N] = f1; 2;:::; Ng ⊂ Z, the combinatorial structures in question are arithmetic progressions of length k (or k-APs), and the assumption on A is just that it has density jAj=N at least α in [N] (for N large enough). It is helpful to rephrase this central result in terms of counting k-APs in A. Notation: given a finite set X and f : X ! C, we define the average of f 1 P on X by Ex2X f (x) = jX j x2X f (x) . Given A ⊂ X we write 1A for the indicator function of A. (1A(x) = 1 for x 2 A and 0 otherwise.) E.g. for A ⊂ [N], Ex2[N]1A(x) is simply jAj=N, the density of A in [N] (also denoted sometimes by P(A)). 2 / 47 An equivalent formulation of Szemer´edi'stheorem If A ⊂ ZN = Z=NZ, we can express the count of k-APs in A in a norma- lized form (i.e. divided by the total number of k-APs in ZN ) as the following average: Ex;d2ZN 1A(x) 1A(x + d) ··· 1A(x + (k − 1)d). Notation: given f1;:::; fk : ZN ! C, let us denote by Sk (f1;:::; fk ) the average Ex;d2ZN f1(x) f2(x + d) ··· fk (x + (k − 1)d). We abbreviate this to Sk (f ) when f1 = ··· = fk = f . It is a known fact (which follows from a result of Varnavides [11]) that Szemer´edi'stheorem has the following equivalent formulation. Theorem Let k 2 N and α > 0. There exists c = c(α; k) > 0 such that for every N 2 N and every f : ZN ! [0; 1] with EZN f ≥ α, we have Sk (f ) ≥ c. As part of a strategy toward proving this theorem, we can search for some useful way to control the value of Sk (f ) in terms of some property of f . 3 / 47 The case k = 3 and Fourier analysis We shall tacitly assume from now on that N is an odd prime. In the 1950s, K.F. Roth made the following observation [8]: given A ⊂ ZN , the average S3(1A) can be controlled in terms of the Fourier transform of 1A, in a way that yields a proof of Szemer´edi'stheorem for k = 3. Basics of discrete Fourier analysis: (let e : R=Z ! C, θ 7! exp(2πiθ)) r x • The characters on ZN are the functions er : ZN ! C, x 7! e( N ), (r 2 ZN ). ∼ r x They form the dual group ZcN = ZN (identify e( N ) with its frequency r). ZN • We equip C with the inner product hf ; gi = Ex2ZN f (x)g(x). The characters form an orthonormal basis of (CZN ; h·; ·i). ZN r x • The Fourier transform of f 2 C is fb : ZcN ! C, r 7! Ex2ZN f (x)e( N ). Thus fb(r) = hf ; er i. Call fb(r) the Fourier coefficient of f at frequency r. • Fourier inversion: for every f 2 ZN , we have f (x) = P f (r) e( r x ). C r2ZN b N p p p 1=p ZN p • We define the L and ` norms on C by kf kL = (Ex2ZN jf (x)j ) , P p 1=p kf k`p = ( jf (x)j ) , p 2 [1; 1). We define also kf k1 = sup jf (x)j. x2ZN x ZN • Plancherel's theorem: for every f 2 C , we have kf kL2 = kfbk`2 . 4 / 47 These foundations of Fourier analysis are not hard to extend to a general finite abelian group Z. In particular Zb =∼ Z. As an explicit isomorphism, we can identify a character χ 2 Zb with some unique frequency r 2 Z, via the r x following notion that generalizes the map (r; x) 7! N we used on ZN . Definition.A bilinear form on Z is a map Z × Z ! R=Z; (r; x) 7! r · x with the following property: for every r 2 Z, the map x 7! r · x is a homo- morphism, and for every x 2 Z the map r 7! r · x is a homomorphism. A bilinear form · is non-degenerate if for every r 2 Z nf0g there is x 2 Z such that r · x 6= 0, and for every x 2 Z nf0g there is r such that r · x 6= 0. We say that the form · is symmetric if r · x = x · r for all r; x 2 Z. Fact [10, x4.1]: every Z has a non-degenerate symmetric bilinear form. Having fixed such a form ·, each χ 2 Zb is a map x 7! e(r · x) for some unique r 2 Z. All the facts on slide 4 then hold similarly on Z. The only explicit examples that we shall use in the course are the following: r x • On ZN , as we saw, r · x = N mod 1 (viewing R=Z as [0; 1) with + mod 1 ). n • Let F be a vector space over a finite field F of prime order p. Then for n 1 r = (r1;:::; rn); x = (x1;:::; xn) 2 F , let r · x = p (r1x1 + ··· + rnxn) mod 1. 5 / 47 The case k = 3 and Fourier analysis The above-mentioned observation of Roth can be stated as follows: Control of the 3-AP-count-deviation using Fourier coefficients 3 P 2 If A ⊂ Z, jAj = αj Z j, j Z j odd, then jS3(1A) − α j ≤ r6=0 j1cA(r)j j1cA(−2r)j. Toward a more general version of this fact: let fA denote the balanced function of A, namely fA(x) = 1A(x) − α (\balanced" since Ex2ZfA(x) = 0). Then S3(1A) = Ex;d2Z (fA + α)(x)(fA + α)(x + d)(fA + α)(x + 2d). 3 A simple calculation shows that this equals α + S3(fA). Thus we have 3 P 2 P 2 S3(1A) − α = S3(fA), and r6=0 1cA(r) 1cA(−2r) = r fbA(r) fbA(−2r) (the latter equality since fbA(0) = EfA = 0 and fbA(r) = 1cA(r) for r 6= 0). Therefore, the above fact is implied by the following result. Proposition (The Fourier transform controls averages over 3-APs) P Let f1; f2; f3 :Z ! C. Then S3(f1; f2; f3) = r2Z fb1(r) fb2(−2r) fb3(r). Moreover, if j Z j is odd and kfj k1 ≤ 1 for all j 2 [3], then jS3(f1; f2; f3)j ≤ minj2[3] kfbj k1. 6 / 47 Proposition (The Fourier transform controls averages over 3-APs) P Let f1; f2; f3 :Z ! C. Then S3(f1; f2; f3) = r2Z fb1(r) fb2(−2r) fb3(r). Moreover, if j Z j is odd and kfj k1 ≤ 1 for all j 2 [3], then jS3(f1; f2; f3)j ≤ minj2[3] kfbj k1. Proof. By Fourier inversion, 8 j 2 [3] we have f (x) = P f (r )e(r · x). j rj 2Z bj j j We substitute this into S3(f1; f2; f3) = Ex;d2Z f1(x) f2(x + d) f3(x + 2d): S (f ; f ; f ) = P f (r ) f (r ) f (r ) er ·x +r ·(x +d)+r ·(x +2d) 3 1 2 3 Ex;d r1;r2;r3 b1 1 b2 2 b3 3 1 2 3 = P f (r ) f (r ) f (r ) ex · (r + r + r ) ed · (r + 2r ). r1;r2;r3 b1 1 b2 2 b3 3 Ex;d 1 2 3 2 3 Using the orthonormality of characters, the inner average here is seen to be Ex e(x ·(r1 +r2 +r3)) Ed e(d ·(r2 +2r3)) = 1(r1 +r2 +r3 = 0) 1(r2 = −2r3) = 1(r1 = r3) 1(r2 = −2r3). We substitute this back into S3 above: P P S3(f1; f2; f3) = r1;r2;r3: fb1(r1) fb2(r2) fb3(r3) = r fb1(r) fb2(−2r) fb3(r). r1=r3; r2=−2r3 To see the second claim, it will suffice to prove it for j = 1. We have P P jS3(f1; f2; f3)j ≤ r jfb1(r)j jfb2(−2r)j jfb3(r)j ≤ kfb1k1 r jfb2(−2r)j jfb3(r)j. P The Cauchy-Schwarz inequality implies r jfb2(−2r)j jfb3(r)j ≤ kfb2k`2 kfb3k`2 . By Plancherel's theorem, this is kf2kL2 kf3kL2 ≤ kf2k1 kf3k1 ≤ 1. 7 / 47 Z It follows from the linearity of the map f 7! fb that kfbk1 is a norm on C , which is often denoted by kf ku2 . Let us restate the proposition: Proposition (The Fourier transform controls averages over 3-APs) Let f1; f2; f3 :Z ! C, kfi k1 ≤ 1, j Z j odd. Then jS3(f1; f2; f3)j ≤ minj2[3] kfj ku2 . This is one of the key analytic facts driving Roth's proof of the case k = 3 of Szemer´edi'stheorem. Indeed, given A ⊂ ZN of density α, applying the 3 proposition to f1 = f2 = f3 = fA yields jα − S3(1A)j ≤ kfAku2 . Hence if 3 2 kfAku2 is small compared to α, then A contains roughly α N 3-APs.
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