A Note on a Problem of Motzkin Regarding Density of Integral Sets with Missing Differences
Ram Krishna Pandey, Patliputra Colony, Amitabha Tripathi · 2011
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)}). 1