Maximal sets with no solution to x+y=3z
Abstract
In this paper, we are interested in a generalization of the notion of sum-free sets. We address a conjecture first made in the 90s by Chung and Goldwasser. Recently, after some computer checks, this conjecture was formulated again by Matolcsi and Ruzsa, who made a first significant step towards it. Here, we prove the full conjecture by giving an optimal upper bound for the Lebesgue measure of a 3-sum-free subset A of [0,1], that is, a set containing no solution to the equation x+y=3z where x,y and z are restricted to belong to A. We then address the inverse problem and characterize precisely, among all sets with that property, those attaining the maximal possible measure.
Cite
@article{arxiv.1211.3341,
title = {Maximal sets with no solution to x+y=3z},
author = {Alain Plagne and Anne de Roton},
journal= {arXiv preprint arXiv:1211.3341},
year = {2013}
}
Comments
Lemma 4 has been corrected. More information is given on the reference [2] in the introduction