Sets Characterized by Missing Sums and Differences in Dilating Polytopes

A sum-dominant set is a finite set A of integers such that |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 sumdominant subsets of {0, . . . , n} is bounded below by a positive constant as n → ∞. Hegarty then extended their work and showed that for any prescribed s, d ∈ N0, the proportion ρ n of subsets of {0, . . . , n} that are missing exactly s sums in {0, . . . , 2n} and exactly 2d differences in {−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 R with vertices in Z , and let ρ n now denote the proportion of subsets of L(nP ) that are missing exactly s sums in L(nP ) + L(nP ) and exactly 2d differences in L(nP ) − L(nP ). As it turns out, the geometry of P has a significant effect on the limiting behavior of ρ n . We define a geometric characteristic of polytopes called local point symmetry, and show that ρ n is bounded below by a positive constant as n → ∞ if and only if P is locally point symmetric. We further show that the proportion of subsets in 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 ) also remains positive in the limit. CONTENTS