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150
Restriction and Kakeya phenomena for finite fields
 DUKE MATH. J
, 2004
"... The restriction and Kakeya problems in Euclidean space have received much attention in the last few decades, and they are related to many problems in harmonic analysis, partial differential equations (PDEs), and number theory. In this paper we initiate the study of these problems on finite fields. I ..."
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Cited by 41 (0 self)
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The restriction and Kakeya problems in Euclidean space have received much attention in the last few decades, and they are related to many problems in harmonic analysis, partial differential equations (PDEs), and number theory. In this paper we initiate the study of these problems on finite fields. In many cases the Euclidean arguments carry over easily to the finite setting (and are, in fact, somewhat cleaner), but there
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.
An improved bound on the Minkowski dimension of Besicovitch sets in R³
 ANNALS OF MATH. 152
, 2000
"... A Besicovitch set is a set which contains a unit line segment in any direction. It is known that the Minkowski and Hausdorff dimensions of such a set must be greater than or equal to 5/2 in R³. In this paper we show that the Minkowski dimension must in fact be greater than 5/2 + ε for some absolute ..."
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Cited by 35 (12 self)
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A Besicovitch set is a set which contains a unit line segment in any direction. It is known that the Minkowski and Hausdorff dimensions of such a set must be greater than or equal to 5/2 in R³. In this paper we show that the Minkowski dimension must in fact be greater than 5/2 + ε for some absolute constant ε> 0. One observation arising from the argument is that Besicovitch sets of nearminimal dimension have to satisfy certain strong properties, which we call “stickiness,” “planiness,” and “graininess.” The purpose of this paper is to improve upon the known bounds for the Minkowski dimension of Besicovitch sets in three dimensions. As a byproduct of the argument we obtain some strong conclusions on the structure of Besicovitch sets with almostminimal Minkowski dimension. Definition 0.1. A Besicovitch set (or “Kakeya set”) E ⊂ Rn is a set which contains a unit line segment in every direction. Informally, the Kakeya conjecture states that all Besicovitch sets in R n
Restriction theory of the Selberg sieve, with applications
 JOURNAL DE THÉORIE DES NOMBRES DE BORDEAUX
, 2005
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BOUNDS ON ARITHMETIC PROJECTIONS, AND APPLICATIONS TO THE KAKEYA CONJECTURE
, 2000
"... Let A, B, be finite subsets of an abelian group, and let G ⊂ A×B be such that #A,#B, #{a + b: (a, b) ∈ G} ≤ N. We consider the question of estimating the quantity #{a − b: (a, b) ∈ G}. In [2] Bourgain obtained ..."
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Cited by 33 (8 self)
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Let A, B, be finite subsets of an abelian group, and let G ⊂ A×B be such that #A,#B, #{a + b: (a, b) ∈ G} ≤ N. We consider the question of estimating the quantity #{a − b: (a, b) ∈ G}. In [2] Bourgain obtained
Bounding multiplicative energy by the sumset
 Adv. Math
"... Abstract. We prove that the sumset or the productset of any finite set of real numbers, A, is at least A  4/3−ε, improving earlier bounds. Our main tool is a new upper bound on the multiplicative energy, E(A, A). 1. ..."
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Abstract. We prove that the sumset or the productset of any finite set of real numbers, A, is at least A  4/3−ε, improving earlier bounds. Our main tool is a new upper bound on the multiplicative energy, E(A, A). 1.
Some connections between Falconer’s distance set conjecture, and sets of Furstenburg type
, 2001
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