A Note on a Problem of Motzkin Regarding Density of Integral Sets with Missing Differences

For a given set M of positive integers, a problem of Motzkin asks to determine the maximal density μ(M) among sets of nonnegative integers in which no two elements differ by an element of M . The problem is completely settled when |M | ≤ 2, and some partial results are known for several families of M when |M | ≥ 3. In 1985 Rabinowitz & Proulx provided a lower bound for μ({a, b, a+ b}) and conjectured that their bound was sharp. Liu & Zhu proved this conjecture in 2004. For each n ≥ 1, we determine κ({a, b, n(a + b)}), which is a lower bound for μ({a, b, n(a + b)}), and conjecture this to be the exact value of μ({a, b, n(a+ b)}).