MSTD sets and Freiman isomorphisms

An MSTD set is a finite set with more pairwise sums than differences. $(\mathrm{{\rm Y}},\mathrm{\Phi })$-ismorphisms are generalizations of Freiman isomorphisms to arbitrary linear forms. These generalized isomorphisms are used to prove that every finite set of real numbers is Freiman isomorphic to a finite set of integers. This implies that there exists no MSTD set $A$ of real numbers with $\mathrm{\mid }A\mathrm{\mid }\le 7$, and, up to Freiman isomorphism, there exists exactly one MSTD set $A$ of real numbers with $\mathrm{\mid }A\mathrm{\mid }=8$.