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.
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Notes
As if often the case when dealing with measure theoretic arguments, it does not make a meaningful difference if we use open or closed intervals (or even half-open/half-closed intervals). Certain arguments are cleaner if one uses one or the other, and thus in this paper we shall sometimes assume that the elements of \(\mathbb {I}\) and \(\mathbb {I}_n\) consist of collections of open intervals, and at other times closed intervals.
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The first named author was partially supported by NSF Grants DMS1265673 and DMS1561945, and the second by DMS1347804. We thank Mel Nathanson, Carsten Sprunger, and Roger Van Peski for helpful conversations, and the referee for a careful reading of the manuscript.
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Miller, S.J., Peterson, C. A Geometric Perspective on the MSTD Question. Discrete Comput Geom 62, 832–855 (2019). https://doi.org/10.1007/s00454-019-00109-7
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DOI: https://doi.org/10.1007/s00454-019-00109-7