Large sets with small doubling modulo p are well covered by an arithmetic progression

Oriol Serra[1]; Gilles Zémor[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)

Annales de l’institut Fourier (2009)

  • Volume: 59, Issue: 5, page 2043-2060
  • ISSN: 0373-0956

Abstract

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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 | 2 S | ≤ ( 2 + ϵ ) | S | and 2 ( | 2 S | ) - 2 | S | + 3 ≤ p is contained in an arithmetic progression of length | 2 S | - | S | + 1 . This is the first result of this nature which places no unnecessary restrictions on the size of S .

How to cite

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Serra, Oriol, and Zémor, Gilles. "Large sets with small doubling modulo $p$ are well covered by an arithmetic progression." Annales de l’institut Fourier 59.5 (2009): 2043-2060. <http://eudml.org/doc/10446>.

@article{Serra2009,
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 $S$.},
affiliation = {Universitat Politècnica de Catalunya Matemàtica Aplicada IV Campus Nord - Edif. C3, C. Jordi Girona, 1-3 08034 Barcelona (Spain); Université Bordeaux 1 Institut de Mathématiques de Bordeaux, UMR 5251 351, cours de la Libération 33405 Talence (France)},
author = {Serra, Oriol, Zémor, Gilles},
journal = {Annales de l’institut Fourier},
keywords = {Sumset; arithmetic progression; additive combinatorics; sumset},
language = {eng},
number = {5},
pages = {2043-2060},
publisher = {Association des Annales de l’institut Fourier},
title = {Large sets with small doubling modulo $p$ are well covered by an arithmetic progression},
url = {http://eudml.org/doc/10446},
volume = {59},
year = {2009},
}

TY - JOUR
AU - Serra, Oriol
AU - Zémor, Gilles
TI - Large sets with small doubling modulo $p$ are well covered by an arithmetic progression
JO - Annales de l’institut Fourier
PY - 2009
PB - Association des Annales de l’institut Fourier
VL - 59
IS - 5
SP - 2043
EP - 2060
AB - 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 $S$.
LA - eng
KW - Sumset; arithmetic progression; additive combinatorics; sumset
UR - http://eudml.org/doc/10446
ER -

References

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