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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, weexpect sum-dominant sets to be rare in some sense. In 2006, however, Martin andO'Bryant showed that the proportion of sum-dominant subsets of $\{0,\dots,n\}$is bounded below by a positive constant as $n\to\infty$. Hegarty then extendedtheir work and showed that for any prescribed $s,d\in\mathbb{N}_0$, theproportion $\rho^{s,d}_n$ of subsets of $\{0,\dots,n\}$ that are missingexactly $s$ sums in $\{0,\dots,2n\}$ and exactly $2d$ differences in$\{-n,\dots,n\}$ also remains positive in the limit.  We consider the following question: are such sets, characterized by theirsums and differences, similarly ubiquitous in higher dimensional spaces? Wegeneralize the integers in a growing interval to the lattice points in adilating polytope. Specifically, let $P$ be a polytope in $\mathbb{R}^D$ withvertices in $\mathbb{Z}^D$, and let $\rho_n^{s,d}$ now denote the proportion ofsubsets of $L(nP)$ that are missing exactly $s$ sums in $L(nP)+L(nP)$ andexactly $2d$ differences in $L(nP)-L(nP)$. As it turns out, the geometry of $P$has a significant effect on the limiting behavior of $\rho_n^{s,d}$. We definea geometric characteristic of polytopes called local point symmetry, and showthat $\rho_n^{s,d}$ is bounded below by a positive constant as $n\to\infty$ ifand only if $P$ is locally point symmetric. We further show that the proportionof 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 pointsymmetric, the proportion of sum-dominant subsets of $L(nP)$ also remainspositive in the limit.

Sets Characterized by Missing Sums and Differences in Dilating Polytopes