On sets with small sumset and m-sum-free sets in Z/pZ

The $3k-4$ conjecture in groups $\mathbb{Z}\mathrm{/}p\mathbb{Z}$ for $p$ prime states that if $A$ is a nonempty subset of $\mathbb{Z}\mathrm{/}p\mathbb{Z}$ satisfying $2A\neq $\mathbb{Z}$/p$\mathbb{Z}$$and$|2A|=2|A|+r \leq \min{3|A|-4,;p-r-4}$,then$A$iscoveredbyanarithmeticprogressionofsizeatmost$|A|+r+1$. A theorem of Serra and Z'emor proves the conjecture provided $r\le 0.0001\mathrm{\mid }A\mathrm{\mid }$, without any additional constraint on $\mathrm{\mid }A\mathrm{\mid }$. Subject to the mild additional constraint $\mathrm{\mid }2A\mathrm{\mid }\le 3p\mathrm{/}4$ (which is optimal in a sense explained in the paper), our first main result improves the bound on $r$, allowing $r\le 0.1368\mathrm{\mid }A\mathrm{\mid }$. We also prove a variant which further improves this bound on $r$ provided $A$ is sufficiently dense. We then give several applications. First we apply the above variant to give a new upper bound for the maximal density of $m$-sum-free sets in $\mathbb{Z}\mathrm{/}p\mathbb{Z}$, i.e., sets $A$ having no solution $(x,y,z)\in A^3$ to the equation $x+y=mz$, where $m\ge 3$ is a fixed integer. The previous best upper bound for this maximal density was $1\mathrm{/}3.0001$ (using the Serra-Z'emor Theorem). We improve this to $1\mathrm{/}3.1955$. We also present a construction following an idea of Schoen, which yields a lower bound for this maximal density of the form $1\mathrm{/}8+o(1{)}_{p\to \mathrm{\infty }}$. Another application of our main results concerns sets of the form $\frac{A+A}{A}$ in ${\mathbb{F}}_p$, and we also improve the structural description of large sum-free sets in $\mathbb{Z}\mathrm{/}p\mathbb{Z}$.