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arXiv · 1811.08869

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

Pierre-Yves Bienvenu·François Hennecart·Ilya Shkredov
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Abstract

Let $p$ a large enough prime number. When $A$ is a subset of $\mathbb{F}_p\smallsetminus\{0\}$ of cardinality $|A|> (p+1)/3$, then an application of Cauchy-Davenport Theorem gives $\mathbb{F}_p\smallsetminus\{0\}\subset A(A+A)$. In this note, we improve on this and we show that if $|A|\ge 0.3051 p$ implies $A(A+A)\supseteq\mathbb{F}_p\smallsetminus\{0\}$. In the opposite direction we show that there exists a set $A$ such that $|A| > (1/8+o(1))p$ and $\mathbb{F}_p\smallsetminus\{0\}\not\subseteq A(A+A)$.

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Pierre-Yves Bienvenu, François Hennecart, Ilya Shkredov. 2018-12-18. A note on the set $\boldsymbol{A(A+A)}$. https://doi.org/10.2140/moscow.2019.8.179

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