A note on the set $A(A+A)$

Abstract

Let $p$ be a large enough prime number. When $A$ is a subset of ${\mathbb{F}}_p\backslash \left\{0\right\}$ of cardinality $\left|A\right|>\left(p+1\right)∕3$, then an application of the Cauchy–Davenport theorem gives ${\mathbb{F}}_p\backslash \left\{0\right\}\subset A\left(A+A\right)$. In this note, we improve on this and we show that $\left|A\right|\ge 0.3051p$ implies $A\left(A+A\right)\supseteq {\mathbb{F}}_p\backslash \left\{0\right\}$. In the opposite direction we show that there exists a set $A$ such that $\left|A\right|>\left(\frac{1}{8}+o\left(1\right)\right)p$ and ${\mathbb{F}}_p\backslash \left\{0\right\}⊈A\left(A+A\right)$.