Results 1  10
of
75
Uniform expansion bound for Cayley graphs of SL2(Fp)
 ANN. MATH
"... We prove that Cayley graphs of SL2(Fp) are expanders with respect to the projection of any fixed elements in SL(2,Z) generating a nonelementary subgroup, and with respect to generators chosen at random in SL2(Fp). ..."
Abstract

Cited by 89 (11 self)
 Add to MetaCart
We prove that Cayley graphs of SL2(Fp) are expanders with respect to the projection of any fixed elements in SL(2,Z) generating a nonelementary subgroup, and with respect to generators chosen at random in SL2(Fp).
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 76 (5 self)
 Add to MetaCart
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.
Linear Degree Extractors and the Inapproximability of MAX CLIQUE and CHROMATIC NUMBER
 THEORY OF COMPUTING
, 2007
"... ... that for all ε> 0, approximating MAX CLIQUE and CHROMATIC NUMBER to within n1−ε are NPhard. We further derandomize results of Khot (FOCS ’01) and show that for some γ> 0, no quasipolynomial time algorithm approximates MAX CLIQUE or CHROMATIC NUMBER to within n/2 (logn)1−γ, unless N˜P = ˜ ..."
Abstract

Cited by 66 (2 self)
 Add to MetaCart
... that for all ε> 0, approximating MAX CLIQUE and CHROMATIC NUMBER to within n1−ε are NPhard. We further derandomize results of Khot (FOCS ’01) and show that for some γ> 0, no quasipolynomial time algorithm approximates MAX CLIQUE or CHROMATIC NUMBER to within n/2 (logn)1−γ, unless N˜P = ˜P. The key to these results is a new construction of dispersers, which are related to randomness extractors. A randomness extractor is an algorithm which extracts randomness from a lowquality random source, using some additional truly random bits. We construct new extractors which require only log2 n + O(1) additional random bits for sources with constant entropy rate, and have constant error. Our dispersers use an arbitrarily small constant
An uncertainty principle for cyclic groups of prime order
, 2003
"... Let G be a finite abelian group, and let f: G → C be a complex function on G. The uncertainty principle asserts that the support supp(f):= {x ∈ G: f(x) ̸ = 0} is related to the support of the Fourier transform ˆ f: G → C by the formula supp(f)supp ( ˆ f)  ≥ G where X  denotes the cardina ..."
Abstract

Cited by 64 (1 self)
 Add to MetaCart
(Show Context)
Let G be a finite abelian group, and let f: G → C be a complex function on G. The uncertainty principle asserts that the support supp(f):= {x ∈ G: f(x) ̸ = 0} is related to the support of the Fourier transform ˆ f: G → C by the formula supp(f)supp ( ˆ f)  ≥ G where X  denotes the cardinality of X. In this note we show that when G is the cyclic group Z/pZ of prime order p, then we may improve this to supp(f)  + supp ( ˆ f)  ≥ p + 1 and show that this is absolutely sharp. As one consequence, we see that a sparse polynomial in Z/pZ consisting of k + 1 monomials can have at most k zeroes. Another consequence is a short proof of the wellknown CauchyDavenport inequality.
Erdős distance problem in vector spaces over finite fields
 Transactions of the American Mathematical Society
"... Abstract. We study the Erdös/Falconer distance problem in vector spaces over finite fields. Let Fq be a finite field with q elements and take E ⊂ F d q, d ≥ 2. We develop a Fourier analytic machinery, analogous to that developed by Mattila in the continuous case, for the study of distance sets in F ..."
Abstract

Cited by 57 (19 self)
 Add to MetaCart
(Show Context)
Abstract. We study the Erdös/Falconer distance problem in vector spaces over finite fields. Let Fq be a finite field with q elements and take E ⊂ F d q, d ≥ 2. We develop a Fourier analytic machinery, analogous to that developed by Mattila in the continuous case, for the study of distance sets in F d q to provide estimates for minimum cardinality of the distance set ∆(E) intermsofthe cardinality of E. Bounds for Gauss and Kloosterman sums play an important role in the proof. 1.
Extracting randomness using few independent sources
 In Proceedings of the 45th Annual IEEE Symposium on Foundations of Computer Science
, 2004
"... In this work we give the first deterministic extractors from a constant number of weak sources whose entropy rate is less than 1/2. Specifically, for every δ> 0 we give an explicit construction for extracting randomness from a constant (depending polynomially on 1/δ) number of distributions over ..."
Abstract

Cited by 50 (6 self)
 Add to MetaCart
(Show Context)
In this work we give the first deterministic extractors from a constant number of weak sources whose entropy rate is less than 1/2. Specifically, for every δ> 0 we give an explicit construction for extracting randomness from a constant (depending polynomially on 1/δ) number of distributions over {0, 1} n, each having minentropy δn. These extractors output n bits, which are 2 −n close to uniform. This construction uses several results from additive number theory, and in particular a recent one by Bourgain, Katz and Tao [BKT03] and of Konyagin [Kon03]. We also consider the related problem of constructing randomness dispersers. For any constant output length m, our dispersers use a constant number of identical distributions, each with minentropy Ω(log n) and outputs every possible mbit string with positive probability. The main tool we use is a variant of the “steppingup lemma ” used in establishing lower bound
Extractors for a constant number of polynomially small minentropy independent sources
 In Proceedings of the 38th Annual ACM Symposium on Theory of Computing
, 2006
"... We consider the problem of randomness extraction from independent sources. We construct an extractor that can extract from a constant number of independent sources of length n, each of which have minentropy n γ for an arbitrarily small constant γ> 0. Our extractor is obtained by composing seeded ..."
Abstract

Cited by 42 (9 self)
 Add to MetaCart
We consider the problem of randomness extraction from independent sources. We construct an extractor that can extract from a constant number of independent sources of length n, each of which have minentropy n γ for an arbitrarily small constant γ> 0. Our extractor is obtained by composing seeded extractors in simple ways. We introduce a new technique to condense independent somewhererandom sources which looks like a useful way to manipulate independent sources. Our techniques are different from those used in recent work [BIW04, BKS + 05, Raz05, Bou05] for this problem in the sense that they do not rely on any results from additive number theory. Using Bourgain’s extractor [Bou05] as a black box, we obtain a new extractor for 2 independent blocksources with few blocks, even when the minentropy is as small as polylog(n). We also show how to modify the 2 source disperser for linear minentropy of Barak et al. [BKS + 05] and the 3 source extractor of Raz [Raz05] to get dispersers/extractors with exponentially small error and linear output length where previously both were constant. In terms of Ramsey Hypergraphs, for every constant 1> γ> 0 our construction gives a family of explicit O(1/γ)uniform hypergraphs on N vertices that avoid cliques and independent sets of (log N)γ size 2.
Affine linear sieve, expanders, and sumproduct
"... This paper is concerned with the following general problem. For j = 1, 2,...,k let Aj be invertible integer coefficient polynomial maps of Z n to Z n (here n ≥ 1 and the inverses of Aj’s are assumed to be of the same type). Let Λ be the group generated by A1,...,Ak and ..."
Abstract

Cited by 42 (8 self)
 Add to MetaCart
(Show Context)
This paper is concerned with the following general problem. For j = 1, 2,...,k let Aj be invertible integer coefficient polynomial maps of Z n to Z n (here n ≥ 1 and the inverses of Aj’s are assumed to be of the same type). Let Λ be the group generated by A1,...,Ak and
Simulating Independence: New Constructions of Condensers, Ramsey Graphs, Dispersers, and Extractors
 In Proceedings of the 37th Annual ACM Symposium on Theory of Computing
, 2005
"... We present new explicit constructions of deterministic randomness extractors, dispersers and related objects. More precisely, a distribution X over binary strings of length n is called a δsource if it assigns probability at most 2 −δn to any string of length n, and for any δ> 0 we construct the ..."
Abstract

Cited by 41 (10 self)
 Add to MetaCart
(Show Context)
We present new explicit constructions of deterministic randomness extractors, dispersers and related objects. More precisely, a distribution X over binary strings of length n is called a δsource if it assigns probability at most 2 −δn to any string of length n, and for any δ> 0 we construct the following poly(n)time computable functions: 2source disperser: D: ({0, 1} n) 2 → {0, 1} such that for any two independent δsources X1, X2 we have that the support of D(X1, X2) is {0, 1}. Bipartite Ramsey graph: Let N = 2 n. A corollary is that the function D is a 2coloring of the edges of KN,N (the complete bipartite graph over two sets of N vertices) such that any induced subgraph of size N δ by N δ is not monochromatic. 3source extractor: E: ({0, 1} n) 2 → {0, 1} such that for any three independent δsources X1, X2, X3 we have that E(X1, X2, X3) is (o(1)close to being) an unbiased random bit. No previous explicit construction was known for either of these, for any δ < 1/2 and these results constitute major progress to longstanding open problems. A component in these results is a new construction of condensers that may be of independent
2source dispersers for subpolynomial entropy and Ramsey graphs beating the FranklWilson construction
 Proceedings of STOC06
, 2006
"... The main result of this paper is an explicit disperser for two independent sources on n bits, each of entropy k = n o(1). Put differently, setting N = 2 n and K = 2 k, we construct explicit N × N Boolean matrices for which no K × K submatrix is monochromatic. Viewed as adjacency matrices of bipartit ..."
Abstract

Cited by 28 (6 self)
 Add to MetaCart
(Show Context)
The main result of this paper is an explicit disperser for two independent sources on n bits, each of entropy k = n o(1). Put differently, setting N = 2 n and K = 2 k, we construct explicit N × N Boolean matrices for which no K × K submatrix is monochromatic. Viewed as adjacency matrices of bipartite graphs, this gives an explicit construction of KRamsey bipartite graphs of size N. This greatly improves the previous bound of k = o(n) of Barak, Kindler, Shaltiel, Sudakov and Wigderson [4]. It also significantly improves the 25year record of k = Õ( √ n) on the special case of Ramsey graphs, due to Frankl and Wilson [9]. The construction uses (besides ”classical ” extractor ideas) almost all of the machinery developed in the last couple of years for extraction from independent sources, including: • Bourgain’s extractor for 2 independent sources of some entropy rate < 1/2 [5] • Raz’s extractor for 2 independent sources, one of which has any entropy rate> 1/2 [18] • Rao’s extractor for 2 independent blocksources of entropy n Ω(1) [17]