Abstract. An MSTD set is a finite set with more pairwise sums than differences. (Υ, Φ)-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 |A| ≤ 7, and, up to Freiman isomorphism and affine isomorphism, there exists exactly one MSTD set A of real numbers with |A| = 8.
Sums and differencesFor every nonempty subset A of an additive abelian group, we define the sumsetand the difference setThe observation that 3 + 2 = 2 + 3 but 3 − 2 = 2 − 3 suggests the reasonable conjecture that a finite set of integers or real numbers should have more differences, or at least as many differences, as sums, but this conjecture is false. For example, the set We have |A − A| = 25 < 26 = |A + A|. Sets with more sums than differences are called MSTD sets.As expected, MSTD sets of integers are rare (e.g. Hegarty and Miller [3], Martin and O'Bryant [5], Zhao [12,13] [7], and Nathanson [9,10]), but there is no adequate classification.If A is an MSTD set, then every affine image of A, that is, every set of the form λ * A + µ = {λa + µ : a ∈ A}