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and T.Tao. The distribution of polynomials over finite fields, with applications to the Gowers norms. submitted
, 2007
"... Abstract. 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 ..."
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Cited by 20 (2 self)
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Abstract. 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. 1.
The inverse conjecture for the Gowers norm over finite fields via the correspondence principle
 Analysis & PDE
"... Abstract. 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 ‖ U d (V) if and only if it correlates with a phase polynomial φ = eF(P) of degree at most d − 1, t ..."
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Cited by 13 (3 self)
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Abstract. 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 ‖ U d (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 and the authors in [2]. 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 in [15], [13] in this setting can be avoided by a slight reformulation of the conjecture. 1.
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 9 (0 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
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 9 (0 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].
TESTING LINEARINVARIANT NONLINEAR PROPERTIES
"... We consider the task of testing properties of Boolean functions that are invariant under linear transformations of the Boolean cube. Previous work in property testing, including the linearity test and the test for ReedMuller codes, has mostly focused on such tasks for linear properties. The one ex ..."
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Cited by 6 (2 self)
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We consider the task of testing properties of Boolean functions that are invariant under linear transformations of the Boolean cube. Previous work in property testing, including the linearity test and the test for ReedMuller codes, has mostly focused on such tasks for linear properties. The one exception is a test due to Green for “triangle freeness”: A function f: F n 2 → F2 satisfies this property if f(x), f(y), f(x + y) do not all equal 1, for any pair x, y ∈ F n 2. Here we extend this test to a more systematic study of testing for linearinvariant nonlinear properties. We consider properties that are described by a single forbidden pattern (and its linear transformations), i.e., a property is given by k points v1,..., vk ∈ F k 2 and f: F n 2 → F2 satisfies the property that if for all linear maps L: F k 2 → F n 2 it is the case that f(L(v1)),..., f(L(vk)) do not all equal 1. We show that this property is testable if the underlying matroid specified by v1,..., vk is a graphic matroid. This extends Green’s result to an infinite class of new properties. Our techniques extend those of Green and in particular we establish a link between the notion of “1complexity linear systems” of Green and Tao, and graphic matroids, to derive the results.
Powers of Sequences and Recurrence
, 2008
"... We study recurrence, and multiple recurrence, properties along the kth powers of a given set of integers. We show that the property of recurrence for some given values of k does not give any constraint on the recurrence for the other powers. This is motivated by similar results in number theory co ..."
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Cited by 5 (5 self)
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We study recurrence, and multiple recurrence, properties along the kth powers of a given set of integers. We show that the property of recurrence for some given values of k does not give any constraint on the recurrence for the other powers. This is motivated by similar results in number theory concerning additive basis of natural numbers. Moreover, motivated by a result of Kamae and MendèsFrance, that links single recurrence with uniform distribution properties of sequences, we look for an analogous result dealing with higher order recurrence and make a related conjecture.
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 2 (1 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.
Simultaneous prime specializations of polynomials over finite fields
"... Recently the author proposed a uniform analogue of the BatemanHorn conjectures for polynomials with coefficients from a finite field (i.e., for polynomials in Fq[T] rather than Z[T]). Here we use an explicit form of the Chebotarev density theorem over function fields to prove this conjecture in par ..."
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Cited by 2 (0 self)
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Recently the author proposed a uniform analogue of the BatemanHorn conjectures for polynomials with coefficients from a finite field (i.e., for polynomials in Fq[T] rather than Z[T]). Here we use an explicit form of the Chebotarev density theorem over function fields to prove this conjecture in particular ranges of the parameters. We give some applications including the solution of a problem posed by C. Hall.
Averages of Euler products, distribution of singular series and the ubiquity of Poisson distribution.
, 2008
"... We discuss in some detail the general problem of computing averages of convergent Euler products, and apply this to examples arising from singular series for the ktuple conjecture and more general problems of polynomial representation of primes. We show that the “singular series” for the ktuple c ..."
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Cited by 2 (1 self)
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We discuss in some detail the general problem of computing averages of convergent Euler products, and apply this to examples arising from singular series for the ktuple conjecture and more general problems of polynomial representation of primes. We show that the “singular series” for the ktuple conjecture have a limiting distribution when taken over ktuples with (distinct) entries of growing size. We also give conditional arguments that would imply that the number of twin primes (or more general polynomial prime patterns) in suitable short intervals are asymptotically Poisson distributed.
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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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.