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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.
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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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.
On the finite field Kakeya problem in two dimensions, preprint
"... Abstract. A twodimensional Besicovitch set over a finite field is a subset of the finite plane containing a line in each direction. In this paper, we conjecture a sharp lower bound for the size of such a subset and prove some results toward this conjecture. 1. ..."
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Abstract. A twodimensional Besicovitch set over a finite field is a subset of the finite plane containing a line in each direction. In this paper, we conjecture a sharp lower bound for the size of such a subset and prove some results toward this conjecture. 1.
On the size of Nikodym sets in finite fields
"... Abstract. Let F denote a finite field of q elements. Define a Nikodym set in F n to be a set B ⊂ F n such that for each x ∈ B c, there exists a line L such that L ∩ B c = {x}. In this note we show that the size of every Nikodym set is at least Cn · q n, where Cn depends only on n. 1. ..."
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Abstract. Let F denote a finite field of q elements. Define a Nikodym set in F n to be a set B ⊂ F n such that for each x ∈ B c, there exists a line L such that L ∩ B c = {x}. In this note we show that the size of every Nikodym set is at least Cn · q n, where Cn depends only on n. 1.
From randomness extraction to rotating needles
 SIGACT News
"... The finite field Kakeya problem deals with the way lines in different directions can overlap in a vector space over a finite field. This problem came up in the study of certain Euclidean problems and, independently, in the search for explicit randomness extractors. We survey recent progress on this ..."
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The finite field Kakeya problem deals with the way lines in different directions can overlap in a vector space over a finite field. This problem came up in the study of certain Euclidean problems and, independently, in the search for explicit randomness extractors. We survey recent progress on this problem and describe several of its applications. 1
KAKEYA SETS, NEW MERGERS AND OLD EXTRACTORS
"... Abstract. A merger is a probabilistic procedure which extracts the randomness out of any (arbitrarily correlated) set of random variables, as long as one of them is uniform. Our main result is an efficient, simple, optimal (to constant factors) merger, which, for k random variables on n bits each, u ..."
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Abstract. A merger is a probabilistic procedure which extracts the randomness out of any (arbitrarily correlated) set of random variables, as long as one of them is uniform. Our main result is an efficient, simple, optimal (to constant factors) merger, which, for k random variables on n bits each, uses a O(log(nk)) seed, and whose error is 1/nk. Our merger can be viewed as a derandomized version of the merger of Lu, Reingold, Vadhan and Wigderson (2003). Its analysis generalizes the recent resolution of the Kakeya problem in finite fields of Dvir (2008). Following the plan set forth by TaShma (1996), who defined mergers as part of this plan, our merger provides the last “missing link ” to a simple and modular construction of extractors for all entropies, which is optimal to constant factors in all parameters. This complements the elegant construction of such extractors given by Guruswami, Umans and Vadhan (2007). We also give simple extensions of our merger in two directions. First, we generalize it to handle the case where no source is uniform – in that case the merger will extract the entropy present in the most random of the given sources. Second, we observe that the merger works just as well in the computational setting, when the sources are efficiently samplable, and computational notions of entropy replace the information theoretic ones.