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Oriol Serra; Gilles Zémor
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, 59 no. 5 (2009), p. 2043-2060, doi: 10.5802/aif.2482
Article: subscription required (your ip address: 50.16.36.153) | Reviews MR 2573196 | Zbl pre05641407
Class. Math.: 11P70
Keywords: Sumset, arithmetic progression, additive combinatorics
[1] Y. F. Bilu, V. F. Lev & I. Z. Ruzsa, “Rectification principles in additive number theory”, Discrete Comput. Geom. 19 (1998) no. 3, Special Issue, p. 343-353, Dedicated to the memory of Paul Erdős
Article | MR 1608875 | Zbl 0899.11002
[2] G. A. Freĭman, “The addition of finite sets. I”, Izv. Vysš. Učebn. Zaved. Matematika 1959 (1959) no. 6 (13), p. 202-213 MR 126388 | Zbl 0096.25904
[3] G. A. Freĭman, “Inverse problems in additive number theory. Addition of sets of residues modulo a prime”, Dokl. Akad. Nauk SSSR 141 (1961), p. 571-573 MR 155810 | Zbl 0109.27203
[4] G. A. Freĭman, Foundations of a structural theory of set addition, American Mathematical Society, 1973, Translated from the Russian, Translations of Mathematical Monographs, Vol 37 MR 360496 | Zbl 0271.10044
[5] Ben Green & Imre Z. Ruzsa, “Sets with small sumset and rectification”, Bull. London Math. Soc. 38 (2006) no. 1, p. 43-52
Article | MR 2201602 | Zbl 1155.11307
[6] Yahya O. Hamidoune, “On the connectivity of Cayley digraphs”, European J. Combin. 5 (1984) no. 4, p. 309-312 MR 782052 | Zbl 0561.05028
[7] Yahya O. Hamidoune, “An isoperimetric method in additive theory”, J. Algebra 179 (1996) no. 2, p. 622-630
Article | MR 1367866 | Zbl 0842.20029
[8] Yahya O. Hamidoune, “Subsets with small sums in abelian groups. I. The Vosper property”, European J. Combin. 18 (1997) no. 5, p. 541-556
Article | MR 1455186 | Zbl 0883.05065
[9] Yahya O. Hamidoune, “Some results in additive number theory. I. The critical pair theory”, Acta Arith. 96 (2000) no. 2, p. 97-119
Article | MR 1814447 | Zbl 0985.11011
[10] Yahya O. Hamidoune & Øystein J. Rødseth, “An inverse theorem mod $p$”, Acta Arith. 92 (2000) no. 3, p. 251-262
Article | Zbl 0945.11003
[11] Yahya O. Hamidoune, Oriol Serra & Gilles Zémor, “On the critical pair theory in $\mathbb{Z}/p\mathbb{Z}$”, Acta Arith. 121 (2006) no. 2, p. 99-115
Article | MR 2216136 | Zbl 1147.11060
[12] Yahya O. Hamidoune, Oriol Serra & Gilles Zémor, “On the critical pair theory in abelian groups: beyond Chowla’s theorem”, Combinatorica 28 (2008) no. 4, p. 441-467
Article | MR 2452844
[13] Vsevolod F. Lev & Pavel Y. Smeliansky, “On addition of two distinct sets of integers”, Acta Arith. 70 (1995) no. 1, p. 85-91
Article | MR 1318763 | Zbl 0817.11005
[14] Melvyn B. Nathanson, Additive number theory, Graduate Texts in Mathematics 165, Springer-Verlag, 1996, Inverse problems and the geometry of sumsets MR 1477155 | Zbl 0859.11002
[15] Øystein J. Rødseth, “On Freiman’s 2.4-Theorem”, Skr. K. Nor. Vidensk. Selsk. (2006) no. 4, p. 11-18 Zbl 1162.11010
[16] Imre Z. Ruzsa, “An application of graph theory to additive number theory”, Sci. Ser. A Math. Sci. (N.S.) 3 (1989), p. 97-109 MR 2314377 | Zbl 0743.05052
[17] Oriol Serra & Gilles Zémor, “On a generalization of a theorem by Vosper”, Integers (2000) MR 1771980 | Zbl 0953.11031
[18] Terence Tao & Van Vu, Additive combinatorics, Cambridge Studies in Advanced Mathematics 105, Cambridge University Press, 2006 MR 2289012 | Zbl 1127.11002
[19] A. G. Vosper, “The critical pairs of subsets of a group of prime order”, J. London Math. Soc. 31 (1956), p. 200-205
Article | MR 77555 | Zbl 0072.03402
Oriol Serra; Gilles Zémor
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, 59 no. 5 (2009), p. 2043-2060, doi: 10.5802/aif.2482
Article: subscription required (your ip address: 50.16.36.153) | Reviews MR 2573196 | Zbl pre05641407
Class. Math.: 11P70
Keywords: Sumset, arithmetic progression, additive combinatorics
Résumé - Abstract
We prove that there is a small but fixed positive integer $\epsilon $ such that for every prime $p$ larger than a fixed integer, every subset $S$ of the integers modulo $p$ which satisfies $|2S|\le (2+\epsilon )|S|$ and $2(|2S|)-2|S|+3\le p$ 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 $S$.
Bibliography
Article | MR 1608875 | Zbl 0899.11002
[2] G. A. Freĭman, “The addition of finite sets. I”, Izv. Vysš. Učebn. Zaved. Matematika 1959 (1959) no. 6 (13), p. 202-213 MR 126388 | Zbl 0096.25904
[3] G. A. Freĭman, “Inverse problems in additive number theory. Addition of sets of residues modulo a prime”, Dokl. Akad. Nauk SSSR 141 (1961), p. 571-573 MR 155810 | Zbl 0109.27203
[4] G. A. Freĭman, Foundations of a structural theory of set addition, American Mathematical Society, 1973, Translated from the Russian, Translations of Mathematical Monographs, Vol 37 MR 360496 | Zbl 0271.10044
[5] Ben Green & Imre Z. Ruzsa, “Sets with small sumset and rectification”, Bull. London Math. Soc. 38 (2006) no. 1, p. 43-52
Article | MR 2201602 | Zbl 1155.11307
[6] Yahya O. Hamidoune, “On the connectivity of Cayley digraphs”, European J. Combin. 5 (1984) no. 4, p. 309-312 MR 782052 | Zbl 0561.05028
[7] Yahya O. Hamidoune, “An isoperimetric method in additive theory”, J. Algebra 179 (1996) no. 2, p. 622-630
Article | MR 1367866 | Zbl 0842.20029
[8] Yahya O. Hamidoune, “Subsets with small sums in abelian groups. I. The Vosper property”, European J. Combin. 18 (1997) no. 5, p. 541-556
Article | MR 1455186 | Zbl 0883.05065
[9] Yahya O. Hamidoune, “Some results in additive number theory. I. The critical pair theory”, Acta Arith. 96 (2000) no. 2, p. 97-119
Article | MR 1814447 | Zbl 0985.11011
[10] Yahya O. Hamidoune & Øystein J. Rødseth, “An inverse theorem mod $p$”, Acta Arith. 92 (2000) no. 3, p. 251-262
Article | Zbl 0945.11003
[11] Yahya O. Hamidoune, Oriol Serra & Gilles Zémor, “On the critical pair theory in $\mathbb{Z}/p\mathbb{Z}$”, Acta Arith. 121 (2006) no. 2, p. 99-115
Article | MR 2216136 | Zbl 1147.11060
[12] Yahya O. Hamidoune, Oriol Serra & Gilles Zémor, “On the critical pair theory in abelian groups: beyond Chowla’s theorem”, Combinatorica 28 (2008) no. 4, p. 441-467
Article | MR 2452844
[13] Vsevolod F. Lev & Pavel Y. Smeliansky, “On addition of two distinct sets of integers”, Acta Arith. 70 (1995) no. 1, p. 85-91
Article | MR 1318763 | Zbl 0817.11005
[14] Melvyn B. Nathanson, Additive number theory, Graduate Texts in Mathematics 165, Springer-Verlag, 1996, Inverse problems and the geometry of sumsets MR 1477155 | Zbl 0859.11002
[15] Øystein J. Rødseth, “On Freiman’s 2.4-Theorem”, Skr. K. Nor. Vidensk. Selsk. (2006) no. 4, p. 11-18 Zbl 1162.11010
[16] Imre Z. Ruzsa, “An application of graph theory to additive number theory”, Sci. Ser. A Math. Sci. (N.S.) 3 (1989), p. 97-109 MR 2314377 | Zbl 0743.05052
[17] Oriol Serra & Gilles Zémor, “On a generalization of a theorem by Vosper”, Integers (2000) MR 1771980 | Zbl 0953.11031
[18] Terence Tao & Van Vu, Additive combinatorics, Cambridge Studies in Advanced Mathematics 105, Cambridge University Press, 2006 MR 2289012 | Zbl 1127.11002
[19] A. G. Vosper, “The critical pairs of subsets of a group of prime order”, J. London Math. Soc. 31 (1956), p. 200-205
Article | MR 77555 | Zbl 0072.03402
© Annales de L'Institut Fourier - ISSN (électronique) : 1777-5310