Lacunary matrices

We study unconditional subsequences of the canonical basis e_rc of elementary matrices in the Schatten class S^p. They form the matrix counterpart to Rudin's Lambda(p) sets of integers in Fourier analysis. In the case of p an even integer, we find a sufficient condition in terms of trails on a bipartite graph. We also establish an optimal density condition and present a random construction of bipartite graphs. As a byproduct, we get a new proof for a theorem of Erdos on circuits in graphs.Comment: 14 page

New examples of noncommutative Λ(p) sets

This is a preprint of an article published in the Illinois Journal of Mathematics, vol.47 (2003), issue 4, pp.1063-1078.In this paper, we introduce a certain combinatorial property Z*(k), which is defined for every integer k ≥ 2, and show that every set E ⊂ Z with the property Z*(k) is necessarily a noncommutative Λ (2k) set. In particular, using number theoretic results about the number of solutions to so-called “S-unit equations,” we show that for any finite set Q of prime numbers, EQ is noncommutative Λ(p) for every real number 2 < p < ∞, where EQ is the set of natural numbers whose prime divisors all lie in the set Q

Short Kloosterman Sums for Polynomials over Finite Fields

This is a preprint of an article published in the Canadian Journal of Mathematics, 55 (2003), pp.225-246.We extend to the setting of polynomials over a finite field certain estimates for short Kloosterman sums originally due to Karatsuba. Our estimates are then used to establish some uniformity of distribution results in the ring Fq[x]/M(x) for collections of polynomials either of the form f−1g−1 or of the form f−1g−1 + afg, where f and g are polynomials coprime to M and of very small degree relative to M, and a is an arbitrary polynomial. We also give estimates for short Kloosterman sums where the summation runs over products of two irreducible polynomials of small degree. It is likely that this result can be used to give an improvement of the Brun-Titchmarsh theorem for polynomials over finite fields

Matrix inequalities with applications to the theory of iterated kernels

NOTICE: this is the author's version of a work that was accepted for publication in Linear Algebra and its Applications. Changes resulting from the publishing process, such as peer review, editing, corrections, structural formatting, and other quality control mechanisms may not be reflected in this document. Changes may have been made to this work since it was submitted for publication. A definitive version was subsequently published in Linear Algebra and its Applications, Volume 362 (2003), pp.275-286. doi:10.1016/S0024-3795(02)00517-7. http://www.elsevier.com/locate/laa.For an m × n matrix A with nonnegative real entries, Atkinson, Moran and Watterson proved the inequality s(A)3 ≤ mns(AAtA), where At is the transpose of A, and s(·) is the sum of the entries. We extend this result to finite products of the form AAtAAt . . .A or AAtAAt . . .At and give some applications to the theory of iterated kernels

Fourier analysis, Schur multipliers on SpS^p and non-commutative Λ(p)-sets

This work deals with various questions concerning Fourier multipliers on $L^p$, Schur multipliers on the Schatten class $S^p$ as well as their completely bounded versions when $L^p$ and $S^p$ are viewed as operator spaces. For this purpose we use subsets of ℤ enjoying the non-commutative Λ(p)-property which is a new analytic property much stronger than the classical Λ(p)-property. We start by studying the notion of non-commutative Λ(p)-sets in the general case of an arbitrary discrete group before turning to the group ℤ

On the norm of an idempotent Schur multiplier on the Schatten class

First published in Proceedings of the American Mathematical Society in 2004, published by the American Mathematical Society. © American Mathematical Society.We show that if the norm of an idempotent Schur multiplier on the Schatten class Sp lies sufficiently close to 1, then it is necessarily equal to 1. We also give a simple characterization of those idempotent Schur multipliers on Sp whose norm is 1

Matrix inequalities with applications to the theory of iterated kernels

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Matrix inequalities with applications to the theory of iterated kernels *

Abstract For an m × n matrix A with nonnegative real entries, Atkinson, Moran and Watterson proved the inequality s(A) 3 ≤ mns(AA t A), where A t is the transpose of A, and s(·) is the sum of the entries. We extend this result to finite products of the form AA t AA t . . . A or AA t AA t . . . A t and give some applications to the theory of iterated kernels.

Smooth values of shifted primes in arithmetic progressions

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