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A STEP BEYOND KNESER'S THEOREM FOR ABELIAN FINITE GROUPS

Published online by Cambridge University Press:  28 January 2003

JEAN-MARC DESHOUILLERS  and
GREGORY A. FREIMAN
JEAN-MARC DESHOUILLERS
Affiliation:
Statistique Mathématique et Applications, EA 2961, Université Victor Segalen Bordeaux 2, 33076 Bordeaux Cedex, France and A2X, UMR 5465 CNRS et Université Bordeaux 1. [email protected]
GREGORY A. FREIMAN
Affiliation:
Usha 11, Ramat Aviv, Tel Aviv, Israel and IHÉS, Paris. [email protected]
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Abstract

A precise description of a subset $\mathcal{A}$ of $\mathbb{Z} / n \mathbb{Z}$ satisfying

$$ | \mathcal{A} + \mathcal{A} | \leq 2.04 | \mathcal{A} | $$

is given. Basically, there exists a subgroup $\mathcal{H}$ of $\mathbb{Z} / n \mathbb{Z}$ such that $\mathcal{A}$ is included in an arithmetic progression of $\ell$ cosets modulo $\mathcal{H}$ and

$$(\ell - 1) | \mathcal{H} | \leq | \mathcal{A} + \mathcal{A} | - | \mathcal{A} |.$$

2000 Mathematical Subject Classification: 11B50, 11B83, 20E34.

Keywords

additive number theoryKneser's theoremstructure theory of set additionfinite abelian groups
Type
Research Article
Information
Proceedings of the London Mathematical Society , Volume 86 , Issue 1 , January 2003 , pp. 1 - 28
Copyright
2003 London Mathematical Society

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