Large sets with small doubling modulo p are well covered by an arithmetic progression
[Les grands ensembles d’entiers de petite somme modulo p sont contenus dans des progressions arithmétiques courtes]
Annales de l'Institut Fourier, Tome 59 (2009) no. 5, pp. 2043-2060.

Nous démontrons qu’il existe un entier strictement positif ϵ, petit mais fixé, tel que pour tout nombre premier p plus grand qu’un entier fixé, tout sous-ensemble S des entiers modulo p qui vérifie |2S|(2+ϵ)|S| et 2(|2S|)-2|S|+3p est contenu dans une progression arithmétique de longueur |2S|-|S|+1. Il s’agit du premier résultat de cette nature qui ne contraint pas inutilement le cardinal de S.

We prove that there is a small but fixed positive integer ϵ such that for every prime p larger than a fixed integer, every subset S of the integers modulo p which satisfies |2S|(2+ϵ)|S| and 2(|2S|)-2|S|+3p is contained in an arithmetic progression of length |2S|-|S|+1. This is the first result of this nature which places no unnecessary restrictions on the size of S.

DOI : 10.5802/aif.2482
Classification : 11P70
Keywords: Sumset, arithmetic progression, additive combinatorics
Mot clés : somme de parties, progression arithmétique, combinatoire additive
Serra, Oriol 1 ; Zémor, Gilles 2

1 Universitat Politècnica de Catalunya Matemàtica Aplicada IV Campus Nord - Edif. C3, C. Jordi Girona, 1-3 08034 Barcelona (Spain)
2 Université Bordeaux 1 Institut de Mathématiques de Bordeaux, UMR 5251 351, cours de la Libération 33405 Talence (France)
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Serra, Oriol; Zémor, Gilles. Large sets with small doubling modulo $p$ are well covered by an arithmetic progression. Annales de l'Institut Fourier, Tome 59 (2009) no. 5, pp. 2043-2060. doi : 10.5802/aif.2482. https://aif.centre-mersenne.org/articles/10.5802/aif.2482/

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