Hyper-atoms and the critical pair Theory

Yahya Ould Hamidoune

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arXiv · 0805.3522

Hyper-atoms and the critical pair Theory

Yahya Ould Hamidoune

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Abstract

We introduce the notion of a hyper-atom. One of the main results of this paper is the $\frac{2|G|}3$--Theorem: Let $S$ be a finite generating subset of an abelian group $G$ of order $\ge 2$. Let $T$ be a finite subset of $G$ such that $2\le |S|\le |T|$, $S+T$ is aperiodic, $0\in S\cap T$ and $$ \frac{2|G|+2}3\ge |S+T|= |S|+|T|-1.$$ Let $H$ be a hyper-atom of $S$. Then $S$ and $T$ are $H$--quasi-periodic. Moreover $ϕ(S)$ and $ϕ(T)$ are arithmetic progressions with the same difference, where $ϕ:G\mapsto G/H$ denotes the canonical morphism. This result implies easily the traditional critical pair Theory and its basic stone: Kemperman's Structure Theorem.

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Yahya Ould Hamidoune. 2008-05-22. Hyper-atoms and the critical pair Theory. https://arxiv.org/abs/0805.3522

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