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 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 ∈ N 0 , the proportion ρ s,d 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 D with vertices in Z D , and let ρ s,d 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 s,d n . We define a geometric characteristic of polytopes called local point symmetry, and show that ρ s,d 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. 19 6. Future Directions 21 Appendix A. Number of Pairs of Strictly Antipodal Vertices 22 References

Problems in additive number theory, V: Affinely inequivalent MSTD sets

“…Steve Miller and his students and colleagues have contributed greatly to this subject (cf. [2,3,8,9,12,13,14,30,29,31]).…”

When Sets Can and Cannot Have MSTD Subsets

“…, n} are sum-dominant as n → ∞, and Zhao, who estimated this percentage at around 4.5 • 10 −4 . See [FP,He,HM,Ma,MO,Na1,Na2,Na3,Ru1,Ru2,Zh3] for general overviews, examples, constructions, bounds on percentages and some generalizations, [MOS,MPR,MS,Zh1] for some explicit constructions of infinite families of sum-dominant sets, and [DKMMW,DKMMWW,MV,Zh2] for some extensions to other settings.…”