SETS CHARACTERIZED BY MISSING SUMS AND DIFFERENCES IN DILATING POLYTOPES

ABSTRACT. 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 sum-dominant 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 ρs,dn 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 RD with vertices in Z D, and let ρs,dn 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 ρs,dn. We define a geometric characteristic of polytopes called local point symmetry, and show that ρs,dn is bounded below by a positive constant a