A note on the set $\mathbf{A}(\mathbf{A}+\mathbf{A})$

Abstract

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

10 pages