Results 1  10
of
30
Recent work connected with the Kakeya problem
 Prospects in mathematics (Princeton, NJ
, 1996
"... A Kakeya set in R n is a compact set E ⊂ R n containing a unit line segment in every direction, i.e. ∀e ∈ S n−1 ∃x ∈ R n: x + te ∈ E ∀t ∈ [ − 1 1 ..."
Abstract

Cited by 107 (2 self)
 Add to MetaCart
(Show Context)
A Kakeya set in R n is a compact set E ⊂ R n containing a unit line segment in every direction, i.e. ∀e ∈ S n−1 ∃x ∈ R n: x + te ∈ E ∀t ∈ [ − 1 1
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. ..."
Abstract

Cited by 75 (5 self)
 Add to MetaCart
(Show Context)
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.
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. ..."
Abstract

Cited by 16 (2 self)
 Add to MetaCart
(Show Context)
We survey recent developments on the Kakeya problem.
An incidence bound for kplanes in F n and a planar variant of the Kakeya maximal function, preprint
"... We discuss a planar variant of the Kakeya maximal function in the setting of a vector space over a finite field. Using methods from incidence combinatorics, we demonstrate that the operator is bounded from L p to L q when 1 ≤ p ≤ kn+k+1 k(k+1) and 1 ≤ q ≤ (n − k)p ′. 1 ..."
Abstract

Cited by 4 (0 self)
 Add to MetaCart
(Show Context)
We discuss a planar variant of the Kakeya maximal function in the setting of a vector space over a finite field. Using methods from incidence combinatorics, we demonstrate that the operator is bounded from L p to L q when 1 ≤ p ≤ kn+k+1 k(k+1) and 1 ≤ q ≤ (n − k)p ′. 1
Incidence Theorems and Their Applications
"... We survey recent (and not so recent) results concerning arrangements of lines, points and other geometric objects and the applications these results have in theoretical computer science and combinatorics. The three main types of problems we will discuss are: 1. Counting incidences: Given a set (or s ..."
Abstract

Cited by 2 (0 self)
 Add to MetaCart
(Show Context)
We survey recent (and not so recent) results concerning arrangements of lines, points and other geometric objects and the applications these results have in theoretical computer science and combinatorics. The three main types of problems we will discuss are: 1. Counting incidences: Given a set (or several sets) of geometric objects (lines, points, etc.), what is the maximum number of incidences (or intersections) that can exist between elements in different sets? We will see several results of this type, such as the SzemerediTrotter theorem, over the reals and over finite fields and discuss their applications in combinatorics (e.g., in the recent solution of Guth and Katz to Erdos ’ distance problem) and in computer science (in explicit constructions of multisource extractors). 2. Kakeya type problems: These problems deal with arrangements of lines that point in different directions. The goal is to try and understand to what extent these lines can overlap one another. We will discuss these questions both over the reals and over finite fields and see how they come up in the
Hunter, Cauchy Rabbit, and Optimal Kakeya Sets
"... A planar set that contains a unit segment in every direction is called a Kakeya set. We relate these sets to a game of pursuit on a cycle Zn. A hunter and a rabbit move on the nodes of Zn without seeing each other. At each step, the hunter moves to a neighbouring vertex or stays in place, while the ..."
Abstract

Cited by 1 (1 self)
 Add to MetaCart
(Show Context)
A planar set that contains a unit segment in every direction is called a Kakeya set. We relate these sets to a game of pursuit on a cycle Zn. A hunter and a rabbit move on the nodes of Zn without seeing each other. At each step, the hunter moves to a neighbouring vertex or stays in place, while the rabbit is free to jump to any node. Adler et al (2003) provide strategies for hunter and rabbit that are optimal up to constant factors and achieve probability of capture in the first n steps of order 1 / log n. We show these strategies yield a Kakeya set consisting of 4n triangles with minimal area, (up to constant), namely Θ(1 / log n). As far as we know, this is the first noniterative construction of a boundaryoptimal Kakeya set. Considering the continuum analog of the game yields a construction of a random Kakeya set from two independent standard Brownian motions {B(s) : s ≥ 0} and {W (s) : s ≥ 0}. Let τt: = min{s ≥ 0: B(s) = t}. Then Xt = W (τt) is a Cauchy process, and K: = {(a,Xt + at) : a, t ∈ [0, 1]} is a Kakeya set of zero area. The area of the εneighborhood of K is as small as possible, i.e., almost surely of order Θ(1/  log ε).
THE (d, k) KAKEYA PROBLEM AND ESTIMATES FOR THE XRAY TRANSFORM
, 2007
"... A (d, k) set is a subset of Rd containing a translate of every kdimensional disc of diameter 1. We show that if (1 + √ 2) k−1 + k> d and k ≥ 2, then every (d, k) set has positive Lebesgue measure. This improves a result of Bourgain, who showed that the analogous statement holds when 2k−1 +k ≥ d ..."
Abstract

Cited by 1 (0 self)
 Add to MetaCart
(Show Context)
A (d, k) set is a subset of Rd containing a translate of every kdimensional disc of diameter 1. We show that if (1 + √ 2) k−1 + k> d and k ≥ 2, then every (d, k) set has positive Lebesgue measure. This improves a result of Bourgain, who showed that the analogous statement holds when 2k−1 +k ≥ d and k ≥ 2. We obtain this improvement in two parts. First, we replace Bourgain’s main estimate with a simple recursive maximal operator bound involving mixednorm estimates for the Xray transform. This method allows us to simplify Bourgain’s proof, allows us to obtain improved bounds for the maximal operator associated with (d, k) sets, and demonstrates that improved estimates for (d, k) sets would follow from new bounds for the Xray transform. Second, we adapt arithmeticcombinatorial methods of Katz and Tao to obtain improved bounds for the Xray transform suitable for use with the recursive maximal operator bound.
ON THE EXACT HAUSDORFF DIMENSION OF FURSTENBERGTYPE SETS
, 904
"... Abstract. In this paper we prove some lower bounds on the Hausdorff dimension of sets of Furstenberg type. Moreover, we extend these results to sets of generalized Furstenberg type, associated to doubling dimension functions. With some additional growth conditions on the dimension function, we obtai ..."
Abstract
 Add to MetaCart
(Show Context)
Abstract. In this paper we prove some lower bounds on the Hausdorff dimension of sets of Furstenberg type. Moreover, we extend these results to sets of generalized Furstenberg type, associated to doubling dimension functions. With some additional growth conditions on the dimension function, we obtain a lower bound on the dimension of “zero dimensional ” Furstenberg sets. 1.