On the density of integral sets with missing differences from sets related to arithmetic progressions (Q627623)

Given a set \(M\) of positive integer, a set \(S\) of non-negative integers is said to be an \(M\)-set if \(a\in S\) and \(b\in S\) imply \(a-b\notin M\). In an unpublished problem collection T. S. Motzkin asks for determining the quantity \(\mu(M)=\bar{\delta}(S)\) where \(S\) varies over all \(M\)-sets and \(\bar{\delta}(S)\) stands for the upper asymptotic density. In the case \(|M|\leq2\) the problem was completely solved by \textit{D. G. Cantor} and \textit{B. Gordon} [J. Comb. Theory, Ser. A 14, 281--287 (1973; Zbl 0277.10043)], and there are known partial answers for several families of \(M\) with \(|M|\geq 3\). In the present paper some cases when \(M\) either contains an arithmetic progression or is contained in an arithmetic progression are settled.