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The inverse conjecture for the Gowers norm over finite fields via the correspondence principle. Submitted (2009)

by Terence Tao, Tamar Ziegler
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An inverse theorem for the Gowers Us+1[N ]-norm

by Ben Green, Terence Tao, Tamar Ziegler - Annals of Math
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...22, 36] for related work on multiple averages and nilmanifolds in ergodic theory. (viii) The analogue of GI(s) in finite fields of large characteristic was established by ergodic-theoretic methods in =-=[4, 34]-=-. (ix) A weaker “local” version of the inverse theorem (in which correlation takes place on a subprogression of [N ] of size ∼ N cs) was established by Gowers [11]. This paper provided a good deal of ...

Optimal testing of Reed-Muller codes

by Arnab Bhattacharyya, Grant Schoenebeck, Madhu Sudan, David Zuckerman , 2009
"... We consider the problem of testing if a given function ..."
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We consider the problem of testing if a given function
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...polynomial is Ω(1). Lovett et al. [4] and Green and Tao [3] disproved this conjecture as stated, but a modification of the conjecture remains open, and was recently proven in high characteristic [9], =-=[10]-=-. These conjectures and the Gowers norms have been extremely influential. For example, Green and Tao [11] used the Gowers norms over the integers to prove that the primes contain arbitrarily long arit...

LINEAR FORMS AND QUADRATIC UNIFORMITY FOR FUNCTIONS . . .

by W.T. Gowers, et al.
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LINEAR FORMS AND HIGHER-DEGREE UNIFORMITY FOR FUNCTIONS ON Fnp

by W. T. Gowers, J. Wolf , 2010
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On higher order Fourier analysis

by Balazs Szegedy , 2014
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AN EQUIVALENCE BETWEEN INVERSE SUMSET THEOREMS AND INVERSE CONJECTURES FOR THE U³ NORM

by Ben Green, Terence Tao , 2009
"... We establish a correspondence between inverse sumset theorems (which can be viewed as classifications of approximate (abelian) groups) and inverse theorems for the Gowers norms (which can be viewed as classifications of approximate polynomials). In particular, we show that the inverse sumset theore ..."
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We establish a correspondence between inverse sumset theorems (which can be viewed as classifications of approximate (abelian) groups) and inverse theorems for the Gowers norms (which can be viewed as classifications of approximate polynomials). In particular, we show that the inverse sumset theorems of Freĭman type are equivalent to the known inverse results for the Gowers U 3 norms, and moreover that the conjectured polynomial strengthening of the former is also equivalent to the polynomial strengthening of the latter. We establish this equivalence in two model settings, namely that of the finite field vector spaces Fn 2, and of the cyclic groups Z/NZ. In both cases the argument involves clarifying the structure of certain types of approximate homomorphism.
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...partial higher order analogues, although14 BEN GREEN AND TERENCE TAO the situation here is much less well understood. To illustrate this phenomenon, consider the following result, recently proven in =-=[3, 27]-=-. Here and for the rest of the section we write F := F5 for definiteness, although the same arguments would work for Fp for any fixed prime p � 5. There are definite issues in extremely low characteri...

Testing low complexity affine-invariant properties

by Arnab Bhattacharyya , Eldar Fischer , Shachar Lovett , 2013
"... Abstract Invariance with respect to linear or affine transformations of the domain is arguably the most common symmetry exhibited by natural algebraic properties. In this work, we show that any low complexity affine-invariant property of multivariate functions over finite fields is testable with a ..."
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Abstract Invariance with respect to linear or affine transformations of the domain is arguably the most common symmetry exhibited by natural algebraic properties. In this work, we show that any low complexity affine-invariant property of multivariate functions over finite fields is testable with a constant number of queries. This immediately reproves, for instance, that the Reed-Muller code over F p of degree d < p is testable, with an argument that uses no detailed algebraic information about polynomials, except that low degree is preserved by composition with affine maps. The complexity of an affine-invariant property P refers to the maximum complexity, as defined by Green and Tao (Ann. Math. 2008), of the sets of linear forms used to characterize P. A more precise statement of our main result is that for any fixed prime p ≥ 2 and fixed integer R ≥ 2, any affine-invariant property P of functions f : F n p → [R] is testable, assuming the complexity of the property is less than p. Our proof involves developing analogs of graphtheoretic techniques in an algebraic setting, using tools from higher-order Fourier analysis.
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... will need later on. Defining the decompositions requires us to introduce the Gowers norm first. 2.1 The Gowers Norm We define Gowers norms in the general setting of arbitrary finite Abelian groups. Definition 7 (Gowers norm) Let G be a finite abelian group and f : G → C. For an integer k ≥ 1, the k’th Gowers norm of f , denoted ‖f‖Uk is defined by: ‖f‖2 k Uk = Ex,y1,y2,...,yk∈G ∏ S⊆[k] Ck−|S|f ( x+ ∑ i∈S yi ) where C denotes the complex conjugation operator. Two facts about the Gowers norm will be absolutely crucial in what follows. First is the Gowers Inverse theorem, established by [BTZ10, TZ10]. Throughout, we let e (x) denote the complex number e2πix/p for x ∈ Fp. Theorem 8 (Gowers Inverse Theorem) Given a positive integer d < p, for every δ > 0, there exists ǫ = ǫ8(δ) such that if f : F n p → R satisfies ‖f‖∞ ≤ 1 and ‖f‖Ud+1 ≥ δ, then there exists a polynomial P : Fnp → Fp of degree at most d so that |Ex[f(x) · e (P (x))] |≥ ǫ. The second is a lemma due to Green and Tao [GT10b] based on repeated applications of the Cauchy-Schwarz inequality. Refer to Definition 5 for the term “complexity”. Lemma 9 Let f1, . . . , fm : F n p → [−1, 1]. Let L = {L1, . . . , Lm} be a system of m line...

Every locally characterized affine-invariant property is testable

by Arnab Bhattacharyya, Eldar Fischer, Hamed Hatami, Pooya Hatami, Shachar Lovett , 2013
"... Let F = Fp for any fixed prime p> 2. An affine-invariant property is a property of functions on Fn that is closed under taking affine transformations of the domain. We prove that all affine-invariant properties that have local characterizations are testable. In fact, we give a proximity-oblivious ..."
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Let F = Fp for any fixed prime p&gt; 2. An affine-invariant property is a property of functions on Fn that is closed under taking affine transformations of the domain. We prove that all affine-invariant properties that have local characterizations are testable. In fact, we give a proximity-oblivious test for any such property P, meaning that given an input function f, we make a constant number of queries to f, always accept if f satisfies P, and otherwise reject with probability larger than a positive number that depends only on the distance between f and P. More generally, we show that any affine-invariant property that is closed under taking restrictions to subspaces and has bounded complexity is testable. We also prove that any property that can be described as the property of being decomposable into a known structure of low-degree polynomials is locally characterized and is, hence, testable. For example, whether a function is a product of two degree-d polynomials, whether a function splits into a product of d linear polynomials, and whether a function has low rank are all examples of degree-structural properties and are therefore locally characterized. Our results use a new Gowers inverse theorem by Tao and Ziegler for low characteristic fields that decomposes any polynomial with large Gowers norm into a function of a small number of low-degree non-classical polynomials. We establish a new equidistribution result for high rank non-classical polynomials that drives the proofs of both the testability results and the local characterization of degree-structural properties.
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...degree d is a function P from Fn to Zpk such that the (d + 1)-th order derivative of P is zero. The integer k − 1 is called the “depth” of P . Classical polynomials have depth 0. We use the result of =-=[TZ11]-=- to obtain non-classical polynomials P1, . . . , PC of degree 6 d such that each g(σi) = Γi(P1, . . . , PC) for some function Γi. We return now to the goal of lower-bounding Eq. (2). By a sequence of ...

Higher order Fourier analysis

by Terence Tao
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An inverse theorem for the Gowers U4norm

by Ben Green, Terence Tao, Tamar Ziegler - Glasg. Math. J
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...s = 1 and 2 (see [11]) as well as the truth of analogues of the conjecture in both ergodic theory [18, 28] and in the “finite field model” in which [N ] is replaced by Fn for some small prime field F =-=[1, 26]-=-. It is also known that this conjecture is necessary, in the following sense. Proposition 1.4 (Necessity of inverse conjecture). Suppose that f : [N ] → C is a 1-bounded function, that (F (g(n)Γ))n∈Z ...

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