Some explicit constructions of sets with more sums than differences

Abstract

We present a variety of new results on finite sets A of integers for which the sumset A+A is larger than the difference set A-A, so-called MSTD (more sums than differences) sets. First we show that there is, up to affine transformation, a unique MSTD subset of {\bf Z} of size 8. Secondly, starting from some examples of size 9, we present several new constructions of infinite families of MSTD sets. Thirdly we show that for every fixed ordered pair of non-negative integers (j,k), as n -> \infty a positive proportion of the subsets of {0,1,2,...,n} satisfy |A+A| = (2n+1) - j, |A-A| = (2n+1) - 2k.