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33
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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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.
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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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
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 39 (7 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
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 (13 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
Recent progress on the Kakeya conjecture
 PROCEEDINGS OF THE 6TH INTERNATIONAL CONFERENCE ON HARMONIC ANALYSIS AND PARTIAL DIFFERENTIAL EQUATIONS (EL ESCORIAL
, 2000
"... We survey recent developments on the Kakeya problem. ..."
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We survey recent developments on the Kakeya problem.
A new bound for finite field Besicovitch sets in four dimensions
 Pacific J. Math. 222, no
, 2008
"... Let F be a finite field with characteristic greater than two. Define a Besicovitch set in F 4 to be a set P ⊆ F 4 containing a line in every direction. The Kakeya conjecture asserts that P  ≈ F  4. In [19] it was shown that P  � F  3. In this paper 1 3+ we improve this to P  � F  16. ..."
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Cited by 9 (1 self)
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Let F be a finite field with characteristic greater than two. Define a Besicovitch set in F 4 to be a set P ⊆ F 4 containing a line in every direction. The Kakeya conjecture asserts that P  ≈ F  4. In [19] it was shown that P  � F  3. In this paper 1 3+ we improve this to P  � F  16. On the other hand, we show that the bound of F  3 is sharp if we relax the assumption that the lines point in different directions. One new feature in the argument is the introduction of a small amount of basic algebraic geometry. 1.
An improved analysis of linear mergers
 Comput. Complex
, 2007
"... Abstract. Mergers are procedures that, with the aid of a short random string, transform k (possibly dependent) random sources into a single random source, in a way that ensures that if one of the input sources has minentropy rate δ then the output has minentropy rate close to δ. Mergers were first ..."
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Cited by 6 (3 self)
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Abstract. Mergers are procedures that, with the aid of a short random string, transform k (possibly dependent) random sources into a single random source, in a way that ensures that if one of the input sources has minentropy rate δ then the output has minentropy rate close to δ. Mergers were first introduced by TaShma [28th STOC, pp. 276285, 1996] and have proven to be a very useful tool in explicit constructions of extractors and condensers. In this work we present a new analysis of the merger construction of Lu et al [35th STOC, pp. 602611, 2003]. We prove that the merger’s output is close to a distribution with minentropy rate of at least 6 11δ. We show that the distance from this distribution is polynomially related to the number of additional random bits that were used by the merger (i.e its seed). We are also able to prove a bound of 4 7δ on the minentropy rate at the cost of increasing the statistical error. Both results are improvements to the previous known lower bound of 1 1 2δ (however, in the 2δ result the error decreases exponentially in the length of the seed). To obtain our results we deviate from the usual linear algebra methods that were used by Lu et al and introduce techniques from additive number theory.
Structure Theory of Set Addition
, 2002
"... The object of these notes is to explain a recent proof by Ruzsa of a famous result of Freiman, some significant modifications of Ruzsa’s proof due to Chang, and all the background material necessary to understand these arguments. Freiman’s theorem concerns the structure of sets with small sumset. Le ..."
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The object of these notes is to explain a recent proof by Ruzsa of a famous result of Freiman, some significant modifications of Ruzsa’s proof due to Chang, and all the background material necessary to understand these arguments. Freiman’s theorem concerns the structure of sets with small sumset. Let A be a subset of
On a planar variant of the Kakeya problem
"... Abstract. A Kn 2set is a set of zero Lebesgue measure containing a translate of every plane in an (n −2)–dimensional manifold in Gr(n, 2), where the manifold fulfills a curvature condition. We show that this is a natural class of sets with respect to the Kakeya problem and prove that dimH(E) ≥ 7/2 ..."
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Abstract. A Kn 2set is a set of zero Lebesgue measure containing a translate of every plane in an (n −2)–dimensional manifold in Gr(n, 2), where the manifold fulfills a curvature condition. We show that this is a natural class of sets with respect to the Kakeya problem and prove that dimH(E) ≥ 7/2 for all K4 2sets E. When the underlying field is replaced by C, we get dimH(E) ≥ 7 for all K4 2sets over C, and we construct an example to show that this is sharp. Thus K4 2sets over C do not necessarily have full Hausdorff dimension. 1.