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 : https://doi.org/10.5802/aif.2482
Classification : 11P70
Mots clés : somme de parties, progression arithmétique, combinatoire additive
@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},
     pages = {2043--2060},
     publisher = {Association des Annales de l{\textquoteright}institut Fourier},
     volume = {59},
     number = {5},
     year = {2009},
     doi = {10.5802/aif.2482},
     mrnumber = {2573196},
     language = {en},
     url = {http://www.numdam.org/articles/10.5802/aif.2482/}
}
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://www.numdam.org/articles/10.5802/aif.2482/

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