Sets characterized by missing sums and differences in dilating polytopes

TextA sum-dominant set is a finite set A   of integers such that |A+A|>|A−A||A+A|>|A−A|. As a typical pair of elements contributes one sum and two differences, we expect sum-dominant sets to be rare in some sense. In 2006, however, Martin and O'Bryant showed that the proportion of sum-dominant subsets of {0,…,n}{0,…,n} is bounded below by a positive constant as n→∞n→∞. Hegarty then extended their work and showed that for any prescribed s,d∈N0s,d∈N0, the proportion ρns,d of subsets of {0,…,n}{0,…,n} that are missing exactly s   sums in {0,…,2n}{0,…,2n} and exactly 2d   differences in {−n,…,n}{−n,…,n} also remains positive in the limit.We consider the following question: are such sets, characterized by their sums and differences, similarly ubiquitous in higher dimensional spaces? We generalize the integers in a growing interval to the lattice points in a dilating polytope. Specifically, let P   be a polytope in RDRD with vertices in ZDZD, and let ρns,d now denote the proportion of subsets of L(nP)L(nP) that are missing exactly s   sums in L(nP)+L(nP)L(nP)+L(nP) and exactly 2d   differences in L(nP)−L(nP)L(nP)−L(nP). As it turns out, the geometry of P   has a significant effect on the limiting behavior of ρns,d. We define a geometric characteristic of polytopes called local point symmetry, and show that ρns,d is bounded below by a positive constant as n→∞n→∞ if and only if P   is locally point symmetric. We further show that the proportion of subsets in L(nP)L(nP) that are missing exactly s sums and at least 2d differences remains positive in the limit, independent of the geometry of P. A direct corollary of these results is that if P   is additionally point symmetric, the proportion of sum-dominant subsets of L(nP)L(nP) also remains positive in the limit.VideoFor a video summary of this paper, please visit http://youtu.be/2M8Qg0E0RAc.