Arithmetic progressions in sumsets and L^p-almost-periodicity
Number Theory
2013-02-26 v2 Combinatorics
Abstract
We prove results about the L^p-almost-periodicity of convolutions. One of these follows from a simple but rather general lemma about approximating a sum of functions in L^p, and gives a very short proof of a theorem of Green that if A and B are subsets of {1,...,N} of sizes alpha N and beta N then A+B contains an arithmetic progression of length at least about exp(c (alpha beta log N)^{1/2}). Another almost-periodicity result improves this bound for densities decreasing with N: we show that under the above hypotheses the sumset A+B contains an arithmetic progression of length at least about exp(c (alpha log N/(log(beta^{-1}))^3)^{1/2}).
Cite
@article{arxiv.1103.6000,
title = {Arithmetic progressions in sumsets and L^p-almost-periodicity},
author = {Ernie Croot and Izabella Laba and Olof Sisask},
journal= {arXiv preprint arXiv:1103.6000},
year = {2013}
}
Comments
15 pages; to appear in Combinatorics, Probability and Computing