Hyper-atoms and the critical pair Theory

Details Hyper-atoms and the critical pair Theory Hyper-atoms and the critical pair Theory

2008-05-22

arxiv.org/abs/0805.3522v1

Mathematics

Number Theory

16 pages

Scientific paper

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.

Affiliated with

Also associated with

No associations

Rating

Hyper-atoms and the critical pair Theory does not yet have a rating. At this time, there are no reviews or comments for this scientific paper.

If you have personal experience with Hyper-atoms and the critical pair Theory, we encourage you to share that experience with our LandOfFree.com community. Your opinion is very important and Hyper-atoms and the critical pair Theory will most certainly appreciate the feedback.

Rate now

Comments { 0 }

Profile ID: LFWR-SCP-O-511657