A geometric perspective on the MSTD question

Abstract

A more sums than differences (MSTD) set $A$ is a subset of $\mathbb{Z}$ for which $|A+A| > |A-A|$. Martin and O'Bryant used probabilistic techniques to prove that a non-vanishing proportion of subsets of $\{1, \dots, n\}$ are MSTD as $n \to \infty$. However, to date only a handful of explicit constructions of MSTD sets are known. We study finite collections of disjoint intervals on the real line, $\mathbb{I}$, and explore the MSTD question for such sets, as well as the relation between such sets and MSTD subsets of $\mathbb{Z}$. In particular we show that every finite subset of $\mathbb{Z}$ can be transformed into an element of $\mathbb{I}$ with the same additive behavior. Using tools from discrete geometry, we show that there are no MSTD sets in $\mathbb{I}$ consisting of three or fewer intervals, but there are MSTD sets for four or more intervals. Furthermore, we show how to obtain an infinite parametrized family of MSTD subsets of $\mathbb{Z}$ from a single such set $A$; these sets are parametrized by lattice points satisfying simple congruence relations contained in a polyhedral cone associated to $A$.