Search

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 ∈ N 0 , the proportion ρ n s , d of subsets of { 0 , … , n } that are missing exactly s sums in { 0 , … , 2 n } 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 R D with vertices in Z D , and let ρ n s , d now denote the proportion of subsets of L ( n P ) that are missing exactly s sums in L ( n P ) + L ( n P ) and exactly 2 d differences in L ( n P ) − L ( n P ) . As it turns out, the geometry of P has a significant effect on the limiting behavior of ρ n s , d . We define a geometric characteristic of polytopes called local point symmetry, and show that ρ n s , 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 ( n P ) 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 ( n P ) also remains positive in the limit. Video For a video summary of this paper, please visit http://youtu.be/2M8Qg0E0RAc . [ABSTRACT FROM AUTHOR]