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Some Properties of Lower Level-Sets of Convolutions

Published online by Cambridge University Press:  12 April 2012

ERNIE CROOT
ERNIE CROOT*
Affiliation:
School of Mathematics, Georgia Institute of Technology, 103 Skiles, Atlanta, GA 30332, USA (e-mail: [email protected])
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Abstract

In the present paper we prove a certain lemma about the structure of ‘lower level-sets of convolutions’, which are sets of the form {x ∈ ℤN : 1A * 1A(x) ≤ γ N} or of the form {x ∈ ℤN : 1A(x) < γ N}, where A is a subset of ℤN. One result we prove using this lemma is that if |A| = θ N and |A+A| ≤ (1 − ϵ)N, 0 < ϵ < 1, then this level-set contains an arithmetic progression of length at least Nc, c = c(θ, ϵ, γ) > 0. It is perhaps possible to obtain such a result using Green's arithmetic regularity lemma (in combination with some ideas of Bourgain [6]); however, our method of proof allows us to obtain non-tower-type quantitative dependence between the constant c and the parameters θ and ϵ. For various reasons (discussed in the paper) one might think, wrongly, that such results would only be possible for level-sets involving triple and higher convolutions.

Keywords

Primary 11B30
Type
Paper
Information
Combinatorics, Probability and Computing , Volume 21 , Issue 4 , July 2012 , pp. 515 - 530
Copyright
Copyright © Cambridge University Press 2012

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