Subset sums in abelian groups

TL;DR

This paper establishes lower bounds on the size of subset sums in finite abelian groups, extending Olson's theorem to more general groups and identifying conditions for optimal bounds.

Contribution

It extends Olson's theorem on subset sums from cyclic groups to general finite abelian groups using a generalized Vosper's Theorem.

Findings

Lower bounds on |Sigma(S)| for symmetric subsets in odd order groups

Exact bounds achieved in most cases, within an additive constant of 2

Extension of Olson's theorem to non-cyclic finite abelian groups

Abstract

Denoting by Sigma(S) the set of subset sums of a subset S of a finite abelian group G, we prove that |Sigma(S)| >= |S|(|S|+2)/4-1 whenever S is symmetric, |G| is odd and Sigma(S) is aperiodic. Up to an additive constant of 2 this result is best possible, and we obtain the stronger (exact best possible) bound in almost all cases. We prove similar results in the case |G| is even. Our proof requires us to extend a theorem of Olson on the number of subset sums of anti-symmetric subsets S from the case of Z_p to the case of a general finite abelian group. To do so, we adapt Olson's method using a generalisation of Vosper's Theorem proved by Hamidoune and Plagne.