A Classification of Small MSTD Sets in Arbitrary Fields

A finite set $A$ in an additive abelian group is a More Sums Than Differences (MSTD) set if $\mathrm{\mid }A+A\mathrm{\mid }>\mathrm{\mid }A-A\mathrm{\mid }$. We prove that no MSTD set of size $5$ exists in an additive abelian group. Using a computer program inspired by Hegarty, we then obtain classifications of MSTD sets of sizes $6$, $7$, $8$, and $9$ in arbitrary fields. We also investigate the minimal cardinality of MSTD sets that are multiplicative subgroups of $\mathbb{Z}\mathrm{/}p\mathbb{Z}$.