#### Vol. 8, No. 2, 2019

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

### Pierre-Yves Bienvenu, François Hennecart and Ilya Shkredov

Vol. 8 (2019), No. 2, 179–188
##### Abstract

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

##### Keywords
sum-product estimates, arithmetic combinatorics, finite fields
Primary: 11B75
##### Milestones
Received: 21 November 2018
Revised: 14 December 2018
Accepted: 29 March 2019
Published: 20 May 2019
##### Authors
 Pierre-Yves Bienvenu Université Lyon 1 CNRS, ICJ UMR 5208 Villeurbanne France François Hennecart Université Jean-Monnet CNRS, ICJ UMR 5208 Saint-Étienne France Ilya Shkredov Steklov Mathematical Institute Divison of Algebra and Number Theory Moscow Russia IITP RAS Moscow Russia