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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 ..."
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Cited by 42 (9 self)
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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.
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 ..."
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Cited by 41 (10 self)
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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
Deterministic Extractors For SmallSpace Sources
, 2006
"... We give polynomialtime, deterministic randomness extractors for sources generated in small space, where we model space s sources on {0, 1} n as sources generated by width 2 s branching programs: For every constant δ> 0, we can extract.99δn bits that are exponentially close to uniform (in variati ..."
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Cited by 30 (3 self)
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We give polynomialtime, deterministic randomness extractors for sources generated in small space, where we model space s sources on {0, 1} n as sources generated by width 2 s branching programs: For every constant δ> 0, we can extract.99δn bits that are exponentially close to uniform (in variation distance) from space s sources of minentropy δn, where s = Ω(n). In addition, assuming an efficient deterministic algorithm for finding large primes, there is a constant η> 0 such that for any ζ> n −η, we can extract m = (δ − ζ)n bits that are exponentially close to uniform from space s sources with minentropy δn, where s = Ω(β 3 n). Previously, nothing was known for δ ≤ 1/2, even for space 0. Our results are obtained by a reduction to a new class of sources that we call independentsymbol sources, which generalize both the wellstudied models of independent sources and symbolfixing sources. These sources consist of a string of n independent symbols over a d symbol alphabet with minentropy k. We give deterministic extractors for such sources when k is as small as polylog(n), for small enough d.
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 ..."
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Cited by 28 (6 self)
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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]
An exposition of bourgain’s 2source extractor
, 2007
"... A construction of Bourgain [Bou05] gave the first 2source extractor to break the minentropy rate 1/2 barrier. In this note, we write an exposition of his result, giving a high level way to view his extractor construction. We also include a proof of a generalization of Vazirani’s XOR lemma that see ..."
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Cited by 21 (0 self)
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A construction of Bourgain [Bou05] gave the first 2source extractor to break the minentropy rate 1/2 barrier. In this note, we write an exposition of his result, giving a high level way to view his extractor construction. We also include a proof of a generalization of Vazirani’s XOR lemma that seems interesting in its own right, and an argument (due to Boaz Barak) that shows that any two source extractor with sufficiently small error must be strong.
Two sources are better than one for increasing the Kolmogorov complexity of infinite sequences
, 2007
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Extracting Kolmogorov complexity with applications to dimension zeroone laws
 IN PROCEEDINGS OF THE 33RD INTERNATIONAL COLLOQUIUM ON AUTOMATA, LANGUAGES, AND PROGRAMMING
, 2006
"... We apply recent results on extracting randomness from independent sources to "extract " Kolmogorov complexity. For any ff; ffl? 0, given a string x with K(x) ? ffjxj, we show how to use a constant number of advice bits to efficiently compute another string y, jyj = \Omega (jxj), ..."
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Cited by 19 (2 self)
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We apply recent results on extracting randomness from independent sources to &quot;extract &quot; Kolmogorov complexity. For any ff; ffl? 0, given a string x with K(x) ? ffjxj, we show how to use a constant number of advice bits to efficiently compute another string y, jyj = \Omega (jxj), with K(y) ? (1 \Gamma ffl)jyj. This result holds for both classical and spacebounded Kolmogorov complexity. We use the extraction procedure for spacebounded complexity to establish zeroone laws for polynomialspace strong dimension. Our results include: (i) If Dimpspace(E) ? 0, then Dimpspace(E=O(1)) = 1. (ii) Dim(E=O(1) j ESPACE) is either 0 or 1. (iii) Dim(E=poly j ESPACE) is either 0 or 1. In other words,
Extractors and rank extractors for polynomial sources
 In FOCS ’07
, 2007
"... In this paper we construct explicit deterministic extractors from polynomial sources, which are distributions sampled by low degree multivariate polynomials over finite fields. This naturally generalizes previous work on extraction from affine sources (which are degree 1 polynomials). A direct conse ..."
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Cited by 17 (8 self)
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In this paper we construct explicit deterministic extractors from polynomial sources, which are distributions sampled by low degree multivariate polynomials over finite fields. This naturally generalizes previous work on extraction from affine sources (which are degree 1 polynomials). A direct consequence is a deterministic extractor for distributions sampled by polynomial size arithmetic circuits over exponentially large fields. The steps in our extractor construction, and the tools (mainly from algebraic geometry) that we use for them, are of independent interest: The first step is a construction of rank extractors, which are polynomial mappings which ”extract” the algebraic rank from any system of low degree polynomials. More precisely, for any n polynomials, k of which are algebraically independent, a rank extractor outputs k algebraically independent polynomials of slightly higher degree. The rank extractors we construct are applicable not only over finite fields but also over fields of characteristic zero. The next step is relating algebraic independence to minentropy. We use a theorem of Wooley to show that these parameters are tightly connected. This allows replacing the algebraic assumption on the source (above) by the natural information theoretic one. It also shows that a rank extractor is already a high quality condenser for polynomial sources over polynomially large fields. Finally, to turn the condensers into extractors, we employ a theorem of Bombieri, giving a character sum estimate for polynomials defined over curves. It allows extracting all the randomness (up to a multiplicative constant) from polynomial sources over exponentially large prime fields.
Product theorems in SL2 and SL3
 Journal of the Institute of Mathematics of Jussieu
"... Abstract We study product theorems for matrix spaces. In particular, we prove the following theorems. Theorem 1. For all ε> 0, there is δ> 0 such that if A ⊂ SL3(Z) is a finite set, then either A intersects a coset of a nilpotent subgroup in a set of size at least A1−ε, or A3 > A1+δ. ..."
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Cited by 17 (0 self)
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Abstract We study product theorems for matrix spaces. In particular, we prove the following theorems. Theorem 1. For all ε> 0, there is δ> 0 such that if A ⊂ SL3(Z) is a finite set, then either A intersects a coset of a nilpotent subgroup in a set of size at least A1−ε, or A3 > A1+δ. Theorem 2. Let A be a finite subset of SL2(C). Then either A is contained in a virtually abelian subgroup, or A3 > cA1+δ for some absolute constant δ> 0. Here A3 = {a1a2a3: ai ∈ A, i = 1, 2, 3} is the 3fold product set of A. §0. Introduction. The aim of this paper is to establish product theorems for matrix spaces, in particular SL2(Z) and SL3(Z). Applications to convolution inequalities will appear in a forthcoming paper. Recall first Tits ’ Alternative for linear groups G over a field of characteristic 0:
The sumproduct phenomenon in arbitrary rings
 Cont. to Disc. Math
"... Abstract. The sumproduct phenomenon predicts that a finite set A in a ring R should have either a large sumset A + A or large product set A · A unless it is in some sense “close ” to a finite subring of R. This phenomenon has been analysed intensively for various specific rings, notably the reals R ..."
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Cited by 13 (2 self)
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Abstract. The sumproduct phenomenon predicts that a finite set A in a ring R should have either a large sumset A + A or large product set A · A unless it is in some sense “close ” to a finite subring of R. This phenomenon has been analysed intensively for various specific rings, notably the reals R and cyclic groups Z/qZ. In this paper we consider the problem in arbitrary rings R, which need not be commutative or contain a multiplicative identity. We obtain rigorous formulations of the sumproduct phenomenon in such rings in the case when A encounters few zerodivisors of R. As applications we recover (and generalise) several sumproduct theorems already in the literature. 1.