Large sets with small doubling modulo $p$ are well covered by an arithmetic progression
Annales de l'Institut Fourier, Volume 59 (2009) no. 5, p. 2043-2060

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 $S$.

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 $S$.

DOI : https://doi.org/10.5802/aif.2482
Classification:  11P70
Keywords: Sumset, arithmetic progression, additive combinatorics
@article{AIF_2009__59_5_2043_0,
author = {Serra, Oriol and Z\'emor, Gilles},
title = {Large sets with small doubling modulo $p$ are well covered by an arithmetic progression},
journal = {Annales de l'Institut Fourier},
publisher = {Association des Annales de l'institut Fourier},
volume = {59},
number = {5},
year = {2009},
pages = {2043-2060},
doi = {10.5802/aif.2482},
mrnumber = {2573196},
zbl = {pre05641407},
language = {en},
url = {http://www.numdam.org/item/AIF_2009__59_5_2043_0}
}

Serra, Oriol; Zémor, Gilles. Large sets with small doubling modulo $p$ are well covered by an arithmetic progression. Annales de l'Institut Fourier, Volume 59 (2009) no. 5, pp. 2043-2060. doi : 10.5802/aif.2482. http://www.numdam.org/item/AIF_2009__59_5_2043_0/

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