Zero sum sets in abelian groups

TL;DR

This paper fully characterizes the sizes of zero-sum sets in abelian groups, providing explicit formulas involving the Möbius function, with simplified forms in special cases like elementary abelian groups.

Contribution

It offers a complete determination of zero-sum set cardinalities in abelian groups and introduces a novel proof using two Möbius transforms.

Findings

Distribution of zero-sum set sizes is fully characterized

A complex summation formula involving the Möbius function is provided

Simplified formulas are derived for special cases like elementary abelian groups

Abstract

The distribution of cardinalities of zero-sum sets in abelian groups is completely determined. A complex summation involving the M\"obius function is given for the general abelian group, while in many special cases, including the case of elementary abelian groups, solved earlier by Li and Wan, it has a compact form. The proof involves two different M\"obius transforms, on positive integers and on set partitions.