A step towards the 3k-4 conjecture in Z/pZ and an application to m-sum-free sets

Abstract

The $3k-4$ conjecture in $\Z/p\Z$ states that if $A$ is a nonempty subset of $\Z/p\Z$ satisfying $2A\neq \Z/p\Z$ and $|2A|=2|A|+r \leq \min\{3|A|-4,\;p-r-3\}$, then $A$ is covered by an arithmetic progression of size at most $|A|+r+1$. In this paper we summarize progress made towards this conjecture in a recent joint paper of the same authors. In that paper we prove first that if $|2A|\le (2+\alpha)|A|-3$ for $\alpha\approx 0.136861$ and $|2A|\le 3p/4$,  then $A$ is efficiently covered by an arithmetic progression, as in the conclusion of the conjecture. With a refined argument we prove that we can go up to $\alpha= (\sqrt{33}-5)/4+o_{|A|,p\to\infty}(1)$ at the cost of restricting $|A|\le (p-r)/3$. We then use this to investigate the maximum size of $m$-sum-free sets for $m\ge 3$, i.e., sets $A\subseteq \Z/p\Z$ such that the equation $x+y=mz$ has no solution in $A$. We obtain that for $m$ fixed, $\lim_{p\to\infty}\max\{|A|/p: A\subseteq\Z/p\Z \ m\text{-sum-free}\}\le 1/3.1955$ (previously, the best known upper bound was $1/3.0001$).