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

Pierre-yves Bienvenu 1 François Hennecart 1 Ilya Shkredov 
1 CTN - Combinatoire, théorie des nombres
ICJ - Institut Camille Jordan [Villeurbanne]
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 $|A|\ge 0.3051 p$ then $A(A+A)\supseteq\mathbb{F}_p\smallsetminus\{0\}$. 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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Submitted on : Thursday, March 7, 2019 - 4:22:54 PM
Last modification on : Monday, June 28, 2021 - 2:26:07 PM

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Pierre-yves Bienvenu, François Hennecart, Ilya Shkredov. A note on the set $\boldsymbol{A(A+A)}$. Moscow Journal of Combinatorics and Number Theory, Moscow Institute of Physics and Technology, 2019, 8 (2), pp.179-188. ⟨10.2140/moscow.2019.8.179⟩. ⟨ujm-02060809⟩



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