A New Proof of Kemperman's Theorem

Tomas Boothby; Matt DeVos; Amanda Montejano

A New Proof of Kemperman's Theorem

Combinatorics 2013-03-19 v2

Authors: Tomas Boothby , Matt DeVos , Amanda Montejano

Abstract

Let $G$ be an additive abelian group and let $A,B \subseteq G$ be finite and nonempty. The pair $(A,B)$ is called critical if the sumset $A+B = {a+b \mid$ a \in A $and$ b\in B $}$ satisfies $|A+B| < |A| + |B|$ . Vosper proved a theorem which characterizes all critical pairs in the special case when $|G|$ is prime. Kemperman generalized this by proving a structure theorem for critical pairs in an arbitrary abelian group. Here we give a new proof of Kemperman's Theorem.

Keywords

abelian groups sumsets critical pairs kemperman's theorem vosper's theorem additive combinatorics

Cite

@article{arxiv.1301.0095,
  title  = {A New Proof of Kemperman's Theorem},
  author = {Tomas Boothby and Matt DeVos and Amanda Montejano},
  journal= {arXiv preprint arXiv:1301.0095},
  year   = {2013}
}

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20 pages, 1 figure

A New Proof of Kemperman's Theorem

Abstract

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