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78
The distribution of polynomials over finite fields, with applications to the Gowers norms
, 2007
"... In this paper we investigate the uniform distribution properties of polynomials in many variables and bounded degree over a fixed finite field F of prime order. Our main result is that a polynomial P: F n → F is poorlydistributed only if P is determined by the values of a few polynomials of lower ..."
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Cited by 40 (2 self)
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In this paper we investigate the uniform distribution properties of polynomials in many variables and bounded degree over a fixed finite field F of prime order. Our main result is that a polynomial P: F n → F is poorlydistributed only if P is determined by the values of a few polynomials of lower degree, in which case we say that P has small rank. We give several applications of this result, paying particular attention to consequences for the theory of the socalled Gowers norms. We establish an inverse result for the Gowers U d+1norm of functions of the form f(x) = eF(P(x)), where P: F n → F is a polynomial of degree less than F, showing that this norm can only be large if f correlates with eF(Q(x)) for some polynomial Q: F n → F of degree at most d. The requirement deg(P) < F  cannot be dropped entirely. Indeed, we show the above claim fails in characteristic 2 when d = 3 and deg(P) = 4, showing that the quartic symmetric polynomial S4 in F n 2 has large Gowers U 4norm but does not correlate strongly with any cubic polynomial. This shows that the theory of Gowers norms in low characteristic is not as simple as previously supposed. This counterexample has also been discovered independently by Lovett, Meshulam, and Samorodnitsky [15]. We conclude with sundry other applications of our main result, including a recurrence result and a certain type of nullstellensatz.
The inverse conjecture for the Gowers norm over finite fields via the correspondence principle
, 2010
"... The inverse conjecture for the Gowers norms U d(V) for finitedimensional vector spaces V over a finite field F asserts, roughly speaking, that a bounded function f has large Gowers norm ‖ f ‖Ud (V) if and only if it correlates with a phase polynomial φ = eF(P) of degree at most d−1, thus P:V→F is a ..."
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Cited by 39 (8 self)
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The inverse conjecture for the Gowers norms U d(V) for finitedimensional vector spaces V over a finite field F asserts, roughly speaking, that a bounded function f has large Gowers norm ‖ f ‖Ud (V) if and only if it correlates with a phase polynomial φ = eF(P) of degree at most d−1, thus P:V→F is a polynomial of degree at most d − 1. In this paper, we develop a variant of the Furstenberg correspondence principle which allows us to establish this conjecture in the large characteristic case char F> d from an ergodic theory counterpart, which was recently established by Bergelson, Tao and Ziegler. In low characteristic we obtain a partial result, in which the phase polynomial φ is allowed to be of some larger degree C(d). The full inverse conjecture remains open in low characteristic; the counterexamples found so far in this setting can be avoided by a slight reformulation of the conjecture.
The quantitative behaviour of polynomial orbits on nilmanifolds
, 2007
"... A theorem of Leibman [19] asserts that a polynomial orbit (g(n)Γ)n∈Z on a nilmanifold G/Γ is always equidistributed in a union of closed subnilmanifolds of G/Γ. In this paper we give a quantitative version of Leibman’s result, describing the uniform distribution properties of a finite polynomial o ..."
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Cited by 38 (3 self)
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A theorem of Leibman [19] asserts that a polynomial orbit (g(n)Γ)n∈Z on a nilmanifold G/Γ is always equidistributed in a union of closed subnilmanifolds of G/Γ. In this paper we give a quantitative version of Leibman’s result, describing the uniform distribution properties of a finite polynomial orbit (g(n)Γ) n∈[N] in a nilmanifold. More specifically we show that there is a factorization g = εg ′ γ, where ε(n) is “smooth”, (γ(n)Γ)n∈Z is periodic and “rational”, and (g ′ (n)Γ)n∈P is uniformly distributed (up to a specified error δ) inside some subnilmanifold G ′ /Γ ′ of G/Γ for all sufficiently dense arithmetic progressions P ⊆ [N]. Our bounds are uniform in N and are polynomial in the error tolerance δ. In a subsequent paper [13] we shall use this theorem to establish the Möbius and Nilsequences conjecture from our earlier paper [12].
The Möbius function is strongly orthogonal to nilsequence
 Ann. of Math
"... Abstract We show that the Möbius function µ(n) is strongly asymptotically orthogonal to any polynomial nilsequence (F (g(n)Γ)) n∈N . Here, G is a simplyconnected nilpotent Lie group with a discrete and cocompact subgroup Γ (so G/Γ is a nilmanifold ), g : Z → G is a polynomial sequence, and In part ..."
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Cited by 32 (1 self)
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Abstract We show that the Möbius function µ(n) is strongly asymptotically orthogonal to any polynomial nilsequence (F (g(n)Γ)) n∈N . Here, G is a simplyconnected nilpotent Lie group with a discrete and cocompact subgroup Γ (so G/Γ is a nilmanifold ), g : Z → G is a polynomial sequence, and In particular, this implies the Möbius and Nilsequence conjecture MN(s) from our earlier paper for every positive integer s. This is one of two major ingredients in our programme to establish a large number of cases of the generalised HardyLittlewood conjecture, which predicts how often a collection ψ1, . . . , ψt : Z d → Z of linear forms all take prime values. The proof is a relatively quick application of the results in our recent companion paper. We give some applications of our main theorem. We show, for example, that the Möbius function is uncorrelated with any bracket polynomial such as n √ 3 n √ 2 . We also obtain a result about the distribution of nilsequences (a n xΓ) n∈N as n ranges only over the primes.
Expander graphs in pure and applied mathematics
 Bull. Amer. Math. Soc. (N.S
"... Expander graphs are highly connected sparse finite graphs. They play an important role in computer science as basic building blocks for network constructions, error correcting codes, algorithms and more. In recent years they have started to play an increasing role also in pure mathematics: number th ..."
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Cited by 30 (3 self)
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Expander graphs are highly connected sparse finite graphs. They play an important role in computer science as basic building blocks for network constructions, error correcting codes, algorithms and more. In recent years they have started to play an increasing role also in pure mathematics: number theory, group theory, geometry and more. This expository article describes their constructions and various applications in pure and applied mathematics. This paper is based on notes prepared for the Colloquium Lectures at the
Linear correlations amongst numbers represented by positive definite binary quadratic forms
 Acta Arith
"... Abstract. Let f1,..., ft be positive definite binary quadratic forms, and letRfi(n) = {(x, y) : fi(x, y) = n}  denote the corresponding representation functions. Employing methods developed by Green and Tao, we deduce asymptotics for linear correlations of these representation functions. More pre ..."
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Cited by 13 (7 self)
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Abstract. Let f1,..., ft be positive definite binary quadratic forms, and letRfi(n) = {(x, y) : fi(x, y) = n}  denote the corresponding representation functions. Employing methods developed by Green and Tao, we deduce asymptotics for linear correlations of these representation functions. More precisely, we study the expression En∈K∩[−N,N]d t∏ i=1 Rfi(ψi(n)), where the ψi form a system of affine linear forms no two of which are affinely related, and where K is a convex body. The minor arc analysis builds on the observation that polynomial subsequences of equidistributed nilsequences are still equidistributed, an observation that could be useful in treating the minor arcs of other arithmetic questions. As a very quick application we give asymptotics to the number of simultaneous zeros of certain systems of quadratic equations in 8 or more variables. Contents
RATIONAL POINTS ON PENCILS OF CONICS AND QUADRICS WITH MANY DEGENERATE FIBRES
"... Abstract. For any pencil of conics or higherdimensional quadrics over Q, with all degenerate fibres defined over Q, we show that the Brauer–Manin obstruction controls weak approximation. The proof is based on the Hasse principle and weak approximation for some special intersections of quadrics over ..."
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Cited by 11 (5 self)
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Abstract. For any pencil of conics or higherdimensional quadrics over Q, with all degenerate fibres defined over Q, we show that the Brauer–Manin obstruction controls weak approximation. The proof is based on the Hasse principle and weak approximation for some special intersections of quadrics over Q, which is a consequence of recent advances in additive combinatorics. 1.
AN EQUIVALENCE BETWEEN INVERSE SUMSET THEOREMS AND INVERSE CONJECTURES FOR THE U³ NORM
, 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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Cited by 11 (3 self)
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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.
Testing low complexity affineinvariant properties
 In Khanna [Kha13
"... 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 affineinvariant property of multivariate functions over finite fields is testable with a constant n ..."
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Cited by 8 (3 self)
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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 affineinvariant property of multivariate functions over finite fields is testable with a constant number of queries. This immediately reproves, for instance, that the ReedMuller code over Fp of degree d < p is testable, with an argument that uses no detailed algebraic information about polynomials, except that having low degree is preserved by composition with affine maps. The complexity of an affineinvariant 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 affineinvariant property P of functions f: F n p → [R] is testable, if the complexity of the property is less than p. Our proof involves developing analogs of graphtheoretic techniques in an algebraic setting, using tools from higherorder Fourier analysis. 1