Sets characterized by missing sums and differences in dilating polytopes

Abstract

Text. 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 ρns,d of subsets of {0, . . .. , n} that are missing exactly s sums in {0, . . .. , 2n} and exactly 2. d 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 ℤD, and let ρns,d now denote the proportion of subsets of L(nP) that are missing exactly s sums in L(nP) + L(nP) and exactly 2. d differences in 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→∞ 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 2. d 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. Video. For a video summary of this paper, please visit http://youtu.be/2M8Qg0E0RAc.