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 \(n \rightarrow \infty \). 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.