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49
A sumproduct estimate in finite fields, and applications
"... Abstract. Let A be a subset of a finite field F: = Z/qZ for some prime q. If F  δ < A  < F  1−δ for some δ> 0, then we prove the estimate A + A  + A · A  ≥ c(δ)A  1+ε for some ε = ε(δ)> 0. This is a finite field analogue of a result of [ESz1983]. We then use this estimate to prove a Sze ..."
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Cited by 46 (3 self)
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Abstract. Let A be a subset of a finite field F: = Z/qZ for some prime q. If F  δ < A  < F  1−δ for some δ> 0, then we prove the estimate A + A  + A · A  ≥ c(δ)A  1+ε for some ε = ε(δ)> 0. This is a finite field analogue of a result of [ESz1983]. We then use this estimate to prove a SzemerédiTrotter type theorem in finite fields, and obtain a new estimate for the Erdös distance problem in finite fields, as well as the threedimensional Kakeya problem in finite fields. 1.
A POLYNOMIAL BOUND IN FREIMAN’S THEOREM
 DUKE MATHEMATICAL JOURNAL VOL. 113, NO. 3
, 2002
"... In this paper the following improvement on Freiman’s theorem on set addition is obtained (see Theorems 1 and 2 in Section 1). Let A ⊂ Z be a finite set such that A + A  < αA. Then A is contained in a proper ddimensional progression P, where d ≤ [α − 1] and log(P/A) < Cα 2 (log α) 3. Earlier ..."
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Cited by 44 (2 self)
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In this paper the following improvement on Freiman’s theorem on set addition is obtained (see Theorems 1 and 2 in Section 1). Let A ⊂ Z be a finite set such that A + A  < αA. Then A is contained in a proper ddimensional progression P, where d ≤ [α − 1] and log(P/A) < Cα 2 (log α) 3. Earlier bounds involved exponential dependence in α in the second estimate. Our argument combines I. Ruzsa’s method, which we improve in several places, as well as Y. Bilu’s proof of Freiman’s theorem.
A sharp bilinear restriction estimate for paraboloids, Geom
 Func. Anal
"... Abstract. Recently Wolff [28] obtained a sharp L 2 bilinear restriction theorem for bounded subsets of the cone in general dimension. Here we adapt the argument of Wolff to also handle subsets of “elliptic surfaces ” such as paraboloids. Except for an endpoint, this answers a conjecture of Machedon ..."
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Cited by 40 (7 self)
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Abstract. Recently Wolff [28] obtained a sharp L 2 bilinear restriction theorem for bounded subsets of the cone in general dimension. Here we adapt the argument of Wolff to also handle subsets of “elliptic surfaces ” such as paraboloids. Except for an endpoint, this answers a conjecture of Machedon and Klainerman, and also improves upon the known restriction theory for the paraboloid and sphere.
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 26 (11 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 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 25 (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
On the size of Kakeya sets in finite fields
 J. AMS
, 2008
"... Abstract. A Kakeya set is a subset of � n, where � is a finite field of q elements, that contains a line in every direction. In this paper we show that the size of every Kakeya set is at least Cn · q n, where Cn depends only on n. This answers a question of Wolff [Wol99]. 1. ..."
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Cited by 25 (4 self)
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Abstract. A Kakeya set is a subset of � n, where � is a finite field of q elements, that contains a line in every direction. In this paper we show that the size of every Kakeya set is at least Cn · q n, where Cn depends only on n. This answers a question of Wolff [Wol99]. 1.
From rotating needles to stability of waves: emerging connections between combinatorics, analysis, and pde
 Notices Amer. Math. Soc
, 2000
"... In 1917 S. Kakeya posed the Kakeya needle problem: What is the smallest area required to rotate a unit line segment (a “needle”) by 180 degrees in the plane? Rotating around the midpoint requires ..."
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Cited by 22 (5 self)
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In 1917 S. Kakeya posed the Kakeya needle problem: What is the smallest area required to rotate a unit line segment (a “needle”) by 180 degrees in the plane? Rotating around the midpoint requires
L p improving bounds for averages along curves
 Michael Christ, Department of Mathematics, University of California, Berkeley, CA 947203840, USA
"... Abstract. We establish local (L p, L q) mapping properties for averages on curves. The exponents are sharp except for endpoints. 1. ..."
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Cited by 22 (2 self)
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Abstract. We establish local (L p, L q) mapping properties for averages on curves. The exponents are sharp except for endpoints. 1.