A note on the set A(A + A)

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

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Keywords

sum-product estimates, arithmetic combinatorics, finite fields

Mathematical Subject Classification 2010

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

Vol. 8, No. 2, 2019

Pierre-Yves Bienvenu, François Hennecart and Ilya Shkredov

Abstract

PDF Access Denied

Keywords

Mathematical Subject Classification 2010

Milestones

Authors