Sets characterized by missing sums and differences in dilating polytopes

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 P^(a,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