A new bound for $A(A+A)$ for large sets

Abstract

For $p$ being a large prime number, and $A\subset {\mathbb{F}}_p$ we prove the following: $(i)$ If $A(A+A)$ does not cover all nonzero residues in ${\mathbb{F}}_p$, then $|A|<p/8+o(p)$. $(ii)$ If $A$ is both sum-free and satisfies $A=A^{\ast }$, then $|A|<p/9+o(p)$. $(iii)$ If $|A|\gg \frac{\log \log p}{\sqrt{\log p}}p$, then $|A+A^{\ast }|⩾(1-o(1))\min (2\sqrt{|A|p},p)$. Here the constants $1/8$, $1/9$, and $2$ are the best possible. The proof involves \emph{wrappers}, subsets of a finite abelian group $G$, with which we `wrap' popular values in convolutions $A\ast B$ for dense sets $A,B\subseteq G$. These objects carry some special structural features, making them capable of addressing both additive-combinatorial and enumerative problems.

13 pages