Beyond sum-free sets in the natural numbers

For an interval [1, N] subset of N, sets S subset of [1, N] with the property that |{(x, y) is an element of S-2 : x + y is an element of S}| = 0, known as sum-free sets, have attracted considerable attention. In this paper, we generalize this notion by considering r(S) = |{(x, y) is an element of S-2 : x + y is an element of S}|, and analyze its behaviour as S ranges over the subsets of [1, N]. We obtain a comprehensive description of the spectrum of attainable r-values, constructive existence results and structural characterizations for sets attaining extremal and near-extremal values.