• DOI:10.1007/s00454-019-00109-7
• Corpus ID: 53639055

A Geometric Perspective on the MSTD Question

@article{Miller2019AGP,
  title={A Geometric Perspective on the MSTD Question},
  author={Steven J. Miller and Carsten Peterson},
  journal={Discrete \& Computational Geometry},
  year={2019},
  pages={1-24}
}

• S. Miller, Carsten Peterson
• Published 2 September 2017
• Mathematics
• Discrete & Computational Geometry

A more sums than differences (MSTD) set A is a subset of $$\mathbb {Z}$$Z for which $$|A+A| > |A-A|$$|A+A|>|A-A|. Martin and O’Bryant used probabilistic techniques to prove that a non-vanishing proportion of subsets of $$\{1, \dots , n\}$${1,⋯,n} are MSTD as $$n \rightarrow \infty $$n→∞. 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, $$\mathbb {I}$$I, and explore the MSTD question for such sets, as well as the relation between such sets and MSTD subsets of $$\mathbb {Z}$$Z. In particular we show that every finite subset of $$\mathbb {Z}$$Z can be transformed into an element of $$\mathbb {I}$$I with the same additive behavior. Using tools from discrete geometry, we show that there are no MSTD sets in $$\mathbb {I}$$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 $$\mathbb {Z}$$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.