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.

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.

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.

DOI: 10.5802/aif.2482
Classification: 11P70
Keywords: Sumset, arithmetic progression, additive combinatorics
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, Volume 59 (2009) no. 5, pp. 2043-2060. doi : 10.5802/aif.2482. http://www.numdam.org/articles/10.5802/aif.2482/

[1] Bilu, Y. F.; Lev, V. F.; Ruzsa, I. Z. Rectification principles in additive number theory, Discrete Comput. Geom., Volume 19 (1998) no. 3, Special Issue, pp. 343-353 (Dedicated to the memory of Paul Erdős) | DOI | MR | Zbl

[2] Freĭman, G. A. The addition of finite sets. I, Izv. Vysš. Učebn. Zaved. Matematika, Volume 1959 (1959) no. 6 (13), pp. 202-213 | MR | Zbl

[3] Freĭman, G. A. Inverse problems in additive number theory. Addition of sets of residues modulo a prime, Dokl. Akad. Nauk SSSR, Volume 141 (1961), pp. 571-573 | MR | Zbl

[4] Freĭman, G. A. Foundations of a structural theory of set addition, American Mathematical Society, Providence, R. I., 1973 (Translated from the Russian, Translations of Mathematical Monographs, Vol 37) | MR | Zbl

[5] Green, Ben; Ruzsa, Imre Z. Sets with small sumset and rectification, Bull. London Math. Soc., Volume 38 (2006) no. 1, pp. 43-52 | DOI | MR | Zbl

[6] Hamidoune, Yahya O. On the connectivity of Cayley digraphs, European J. Combin., Volume 5 (1984) no. 4, pp. 309-312 | MR | Zbl

[7] Hamidoune, Yahya O. An isoperimetric method in additive theory, J. Algebra, Volume 179 (1996) no. 2, pp. 622-630 | DOI | MR | Zbl

[8] Hamidoune, Yahya O. Subsets with small sums in abelian groups. I. The Vosper property, European J. Combin., Volume 18 (1997) no. 5, pp. 541-556 | DOI | MR | Zbl

[9] Hamidoune, Yahya O. Some results in additive number theory. I. The critical pair theory, Acta Arith., Volume 96 (2000) no. 2, pp. 97-119 | DOI | MR | Zbl

[10] Hamidoune, Yahya O.; Rødseth, Øystein J. An inverse theorem mod p, Acta Arith., Volume 92 (2000) no. 3, pp. 251-262 | Zbl

[11] Hamidoune, Yahya O.; Serra, Oriol; Zémor, Gilles On the critical pair theory in /p, Acta Arith., Volume 121 (2006) no. 2, pp. 99-115 | DOI | MR | Zbl

[12] Hamidoune, Yahya O.; Serra, Oriol; Zémor, Gilles On the critical pair theory in abelian groups: beyond Chowla’s theorem, Combinatorica, Volume 28 (2008) no. 4, pp. 441-467 | DOI | MR

[13] Lev, Vsevolod F.; Smeliansky, Pavel Y. On addition of two distinct sets of integers, Acta Arith., Volume 70 (1995) no. 1, pp. 85-91 | MR | Zbl

[14] Nathanson, Melvyn B. Additive number theory, Graduate Texts in Mathematics, 165, Springer-Verlag, New York, 1996 (Inverse problems and the geometry of sumsets) | MR | Zbl

[15] Rødseth, Øystein J. On Freiman’s 2.4-Theorem, Skr. K. Nor. Vidensk. Selsk. (2006) no. 4, pp. 11-18 | Zbl

[16] Ruzsa, Imre Z. An application of graph theory to additive number theory, Sci. Ser. A Math. Sci. (N.S.), Volume 3 (1989), pp. 97-109 | MR | Zbl

[17] Serra, Oriol; Zémor, Gilles On a generalization of a theorem by Vosper, Integers (2000), pp. A10, 10 pp. (electronic) | MR | Zbl

[18] Tao, Terence; Vu, Van Additive combinatorics, Cambridge Studies in Advanced Mathematics, 105, Cambridge University Press, Cambridge, 2006 | MR | Zbl

[19] Vosper, A. G. The critical pairs of subsets of a group of prime order, J. London Math. Soc., Volume 31 (1956), pp. 200-205 | DOI | MR | Zbl

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