MSTD sets and Freiman isomorphisms

TL;DR

This paper explores generalized Freiman isomorphisms for linear forms, proving that all finite real sets are isomorphic to integer sets, and characterizing MSTD sets of small size.

Contribution

It introduces $(,)$-ismorphisms as a generalization of Freiman isomorphisms and applies them to classify MSTD sets of real numbers.

Findings

No MSTD set of real numbers with size ≤ 7 exists.

Up to isomorphism, exactly one MSTD set of size 8 exists.

All finite real sets are Freiman isomorphic to integer sets.

Abstract

An MSTD set is a finite set with more pairwise sums than differences. $(\Upsilon,\Phi)$-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 $|A| \leq 7$, and, up to Freiman isomorphism, there exists exactly one MSTD set $A$ of real numbers with $|A| = 8$.