A randomness extractor is an algorithm which extracts randomness from a low-quality random source, using some additional truly random bits. We construct new extractors which require only log n + O(1) additional random bits for sources with constant entropy rate. We further construct dispersers, which are similar to one-sided extractors, which use an arbitrarily small constant times log n additional random bits for sources with constant entropy rate. Our extractors and dispersers output 1-α fraction of the randomness, for any α>0.We use our dispersers to derandomize results of Hastad [23] and Feige-Kilian [19] and show that for all e>0, approximating MAX CLIQUE and CHROMATIC NUMBER to within n1-e are NP-hard. We also derandomize the results of Khot [29] and show that for some γ > 0, no quasi-polynomial time algorithm approximates MAX CLIQUE or CHROMATIC NUMBER to within n/2(log n)1-γ, unless NP = P.Our constructions rely on recent results in additive number theory and extractors by Bourgain-Katz-Tao [11], Barak-Impagliazzo-Wigderson [5], Barak-Kindler-Shaltiel-Sudakov-Wigderson [6], and Raz [36]. We also simplify and slightly strengthen key theorems in the second and third of these papers, and strengthen a related theorem by Bourgain [10].

http://toc.nada.kth.se/articles/v003a006/v003a006.pdf

Linear degree extractors and the inapproximability of max clique and chromatic number

Let Fp be the field of a prime order p and F*p be its multiplicative subgroup In this paper we obtain a variant of sum-product estimates which in particular implies the bound for any subset A ⊂ Fp with 1 p1/4, then Finally we show that if g is a generator of F*p then for any M < p the number of solutions of the equation is less than  This implies that if p1/2 < M < p, then

On a variant of sum-product estimates and explicit exponential sum bounds in prime fields

We prove a pointwise and average bound for the number of incidences between points and hyperplanes in vector spaces over finite fields. While our estimates are, in general, sharp, we observe an improvement for product sets and sets contained in a sphere. We use these incidence bounds to obtain significant improvements on the arithmetic problem of covering F q , the finite field with q elements, by A · A + ··· + A · A, where A is a subset F q of sufficiently large size. We also use the incidence machinery and develop arithmetic constructions to study the Erdos-Falconer distance conjecture in vector spaces over finite fields. We prove that the natural analog of the Euclidean Erdos-Falconer distance conjecture does not hold in this setting. On the positive side, we obtain good exponents for the Erdos-Falconer distance problem for subsets of the unit sphere in F d q and discuss their sharpness. This results in a reasonably complete description of the Erdos-Falconer distance problem in higher-dimensional vector spaces over general finite fields.

https://users.math.msu.edu/users/koh/papers/Tams.pdf

Averages over hyperplanes, sum-product theory in vector spaces over finite fields and the Erdos-Falconer distance conjecture

Let σ and w be locally finite positive Borel measures on R which do not share a common point mass. Assume that the pair of weights satisfy a Poisson A2 condition, and satisfy the testing conditions below, for the Hilbert transform H, ∫IH(σ1I)2dw≲σ(I),∫IH(w1I)2dσ≲w(I), with constants independent of the choice of interval I. Then H(σ⋅) maps L2(σ) to L2(w), verifying a conjecture of Nazarov, Treil, and Volberg. The proof uses basic tools of nonhomogeneous analysis with two components particular to the Hilbert transform. The first component is a global-to-local reduction which is a consequence of prior work by Lacey, Sawyer, Shen, and Uriarte-Tuero. The second component, an analysis of the local part, is the particular contribution of this article.

Two-weight inequality for the hilbert transform: A real variable characterization, I

We prove a point-wise and average bound for the number of incidences between points and hyper-planes in vector spaces over finite fields. While our estimates are, in general, sharp, we observe an improvement for product sets and sets contained in a sphere. We use these incidence bounds to obtain significant improvements on the arithmetic problem of covering ${\mathbb F}_q$, the finite field with q elements, by $A \cdot A+... +A \cdot A$, where A is a subset ${\mathbb F}_q$ of sufficiently large size. We also use the incidence machinery we develope and arithmetic constructions to study the Erdos-Falconer distance conjecture in vector spaces over finite fields. We prove that the natural analog of the Euclidean Erdos-Falconer distance conjecture does not hold in this setting due to the influence of the arithmetic. On the positive side, we obtain good exponents for the Erdos -Falconer distance problem for subsets of the unit sphere in $\mathbb F_q^d$ and discuss their sharpness. This results in a reasonably complete description of the Erdos-Falconer distance problem in higher dimensional vector spaces over general finite fields.

max(\A + A\,\AA\)>Ce\A\?-*. In the finite field setting this situation is much more complicated because the main tool, the Szemer?di-Trotter incidence theorem, does not hold in the same generality. It is known, via the work in [BKT], that if A is a subset of Fp, the field of p elements with p prime, and if p6 0, then one has the sum product estimate max(|? + A|,|AA|)>|,4|1+ for some e > 0. This result has found many applications in combinatorial problems and exponential sum estimates (see e.g. [BKT], [BGK], [G2]). Recently, Garaev [Gl] showed that when \A\ < pz, one has the estimate

/pdf/a-slight-improvement-to-garaev-s-sum-product-estimate-38bnz4fhkz.pdf

A slight improvement to Garaev’s sum product estimate

The two weight inequality for the Hilbert transform arises in the settings of analytic function spaces, operator theory, and spectral theory, and what would be most useful is a characterization in the simplest real-variable terms. We show that the $L^2$ to $L^2$ inequality holds if and only if two L^2 to weak-L^2 inequalities hold. This is a corollary to a characterization in terms of a two-weight Poisson inequality, and a pair of testing inequalities on bounded functions.

Two Weight Inequality for the Hilbert Transform: A Real Variable Characterization, II

Let F
 p
 be the field of a prime order p. For a subset $${A \subset F_p}$$
 we consider the product set A(A + 1). This set is an image of A ×  A under the polynomial mapping f(x, y) = xy +  x : F
 p
 ×  F
 p
 → F
 p
 . In the present note we show that if |A| <  p
 1/2, then

 $$|A(A + 1)| \ge |A|^{106/105+o(1)}.$$
 If |A| >  p
 2/3, then we prove that

 $$|A(A + 1)| \gg \sqrt{p\, |A|}$$
 and show that this is optimal in general settings bound up to the implied constant. We also estimate the cardinality of A(A +  1) when A is a subset of real numbers. We show that in this case one has the Elekes type bound

 $$|A(A + 1)| \gg |A|^{5/4}.$$

On the size of the set A ( A + 1)

In the present paper, we extend Garaev’s techniques to the set of fields which are not necessarily of prime order. Our goal here is just to find an explicit estimate in the supercritical setting where the set A has less cardinality than the square root of the cardinality of the field, and interacts in a less than half-dimensional way with any subfields. (We make this precise below.) Precisely, we

Chun-Yen Shen

Papers

Two-weight inequality for the hilbert transform: A real variable characterization, I

A slight improvement to Garaev’s sum product estimate

Two Weight Inequality for the Hilbert Transform: A Real Variable Characterization, II

On the size of the set A ( A + 1)

Garaev's inequality in Finite Fields not of prime order