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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 ..."
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Cited by 62 (2 self)
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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. ..."
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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.
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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Cited by 10 (2 self)
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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 ..."
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Cited by 4 (0 self)
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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
THE (d, k) KAKEYA PROBLEM AND ESTIMATES FOR THE XRAY TRANSFORM By
, 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 and ..."
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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. Acknowledgements ii I would first like to thank my advisor, Andreas Seeger. Through my years in Madison, he has continuously sought opportunities for me to grow as a mathematician, and he has always looked out for my best interests. I hope to help my future students as much
KAKEYA CONFIGURATIONS IN LIE GROUPS AND HOMOGENEOUS SPACES.
"... Abstract. In this paper, we study continuous Kakeya line and needle configurations, of both the oriented and unoriented varieties, in connected Lie groups and some associated homogenous spaces. These are the analogs of Kakeya line (needle) sets (subsets of Rn where it is possible to turn a line (res ..."
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Abstract. In this paper, we study continuous Kakeya line and needle configurations, of both the oriented and unoriented varieties, in connected Lie groups and some associated homogenous spaces. These are the analogs of Kakeya line (needle) sets (subsets of Rn where it is possible to turn a line (respectively an interval of unit length) through all directions continuously, without repeating a “direction”.) We show under some general assumptions that any such continuous Kakeya line configuration set in a connected Lie group must contain an open neighborhood of the identity, and hence must have positive Haar measure. In connected nilpotent Lie groups G, the only subspace of G that contains such an unoriented line configuration is shown to be G itself. Finally some similar questions in homogeneous spaces are addressed.
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 ..."
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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.
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 ..."
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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
FROM HARMONIC ANALYSIS TO ARITHMETIC COMBINATORICS
, 2007
"... Arithmetic combinatorics, or additive combinatorics, is a fast developing area of research combining elements of number theory, combinatorics, harmonic analysis and ergodic theory. Its arguably bestknown result, and the one that brought it to global prominence, is the proof by Ben Green and Terence ..."
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Arithmetic combinatorics, or additive combinatorics, is a fast developing area of research combining elements of number theory, combinatorics, harmonic analysis and ergodic theory. Its arguably bestknown result, and the one that brought it to global prominence, is the proof by Ben Green and Terence Tao of the longstanding conjecture that primes contain arbitrarily long arithmetic progressions. There are many accounts and expositions of the GreenTao theorem, including the articles by Kra [119] and Tao [182] in the Bulletin. The purpose of the present article is to survey a broader, highly interconnected network of questions and results, built over the decades and spanning several areas of mathematics, of which the GreenTao theorem is a famous descendant. An old geometric problem lies at the heart of key conjectures in harmonic analysis. A major result in partial differential equations invokes combinatorial theorems on intersecting lines and circles. An unexpected argument points harmonic analysts towards additive number theory, with consequences that could have hardly been anticipated.
Lines Missing Every Random Point ∗
"... We prove that there is, in every direction in Euclidean space, a line that misses every computably random point. We also prove that there exist, in every direction in Euclidean space, arbitrarily long line segments missing every double exponential time random point. 1 ..."
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We prove that there is, in every direction in Euclidean space, a line that misses every computably random point. We also prove that there exist, in every direction in Euclidean space, arbitrarily long line segments missing every double exponential time random point. 1