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

TL;DR

This paper establishes new bounds on the size of subsets in finite fields related to their sum and product sets, using novel structural tools called wrappers to address additive combinatorial problems.

Contribution

It introduces the concept of wrappers to analyze sum-product phenomena and derives optimal bounds for large subsets in finite fields.

Findings

If $A(A+A)$ does not cover all nonzero residues, then $|A| < p/8 + o(p)$.

Sum-free sets with $A = A^*$ have size less than $p/9 + o(p)$.

Large sets satisfy $|A + A^*| o ext{almost } 2 ext{sqrt}|A|p$.

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 $A = A^*$, then $|A| < p/9 + o(p)$. $(iii)$ If $|A| \gg \frac{\log\log{p}}{\sqrt{\log{p}}}p$, then $|A + A^*| \geqslant (1 - o(1))\min(2\sqrt{|A|p}, p)$. 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 $A * B$ 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.