Hyper-atoms and the critical pair Theory
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 $\phi(S)$ and $\phi(T)$ are arithmetic progressions with the same difference, where $\phi :G\mapsto G/H$ denotes the canonical morphism. This result implies easily the traditional critical pair Theory and its basic stone: Kemperman's Structure Theorem.
- Publication:
-
arXiv e-prints
- Pub Date:
- May 2008
- DOI:
- 10.48550/arXiv.0805.3522
- arXiv:
- arXiv:0805.3522
- Bibcode:
- 2008arXiv0805.3522O
- Keywords:
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- Mathematics - Number Theory;
- 11B60;
- 11B34;
- 20D60
- E-Print:
- 16 pages