Beyond Sum-Free Sets in the Natural Numbers

Abstract

For an interval $[1,N] \subseteq \mathbb{N}$, sets $S \subseteq [1,N]$ with the property that $|\{(x,y) \in S^2:x+y \in S\}|=0$, known as sum-free sets, have attracted considerable attention.  In this paper, we generalize this notion by considering $r(S)=|\{(x,y) \in S^2: x+y \in S\}|$, 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.