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

For a given set

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

. The problem is completely settled when $\vert M\vert \le 2$ , and some partial results are known for several families of

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 ${\mu}(\{a,b,n(a+b)\})$ .

Received January 6 2011; revised version received May 18 2011. Published in Journal of Integer Sequences, June 1 2011.

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

Ram Krishna Pandey Department of Mathematics Indian Institute of Technology Patliputra Colony, Patna -- 800013 India Amitabha Tripathi Department of Mathematics Indian Institute of Technology Hauz Khas, New Delhi -- 110016 India

Ram Krishna Pandey
Department of Mathematics
Indian Institute of Technology
Patliputra Colony, Patna -- 800013
India

Amitabha Tripathi
Department of Mathematics
Indian Institute of Technology
Hauz Khas, New Delhi -- 110016
India