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 $|2S|\le \left(2+ϵ\right)|S|$ et $2\left(|2S|\right)-2|S|+3\le p$ 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 $|2S|\le \left(2+ϵ\right)|S|$ and $2\left(|2S|\right)-2|S|+3\le p$ 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 : https://doi.org/10.5802/aif.2482
Classification : 11P70
Mots clés : somme de parties, progression arithmétique, combinatoire additive
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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. http://archive.numdam.org/articles/10.5802/aif.2482/

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