Sets with few differences in abelian groups (Q1658765)

Summary: Let \((G, +)\) be an abelian group. In [J. Algebra 287, No. 2, 449--457 (2005; Zbl 1095.11012)] \textit{S. Eliahou} and \textit{M. Kervaire} found an explicit formula for the smallest possible cardinality of the sumset \(A+A\), where \(A \subseteq G\) has fixed cardinality \(r\). We consider instead the smallest possible cardinality of the difference set \(A-A\), which is always greater than or equal to the smallest possible cardinality of \(A+A\) and can be strictly greater. We conjecture a formula for this quantity and prove the conjecture in the case that \(G\) is an elementary abelian \(p\)-group. This resolves a conjecture of Bajnok and Matzke on signed sumsets.