Note on a problem of Motzkin regarding density of integral sets with missing differences

Pandey, Ram Krishna ; Tripathi, Amitabha

Journal of integer sequences, Tome 14 (2011) no. 6

Résumé

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

Détail

Classification : 11B05
Keywords: density, coloring