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 Fp \ {0} of cardinality |A| > (p + 1)3, then an application of the Cauchy–Davenport theorem gives Fp \ {0} A(A + A). In this note, we improve on this and we show that |A| 0.3051p implies A(A + A) Fp \ {0}. In the opposite direction we show that there exists a set A such that |A| >(1 8 + o(1))p and Fp \ {0}A(A + A).

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