Journal of Integer Sequences, Vol. 14 (2011), Article 11.6.3

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

Abstract:

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)\})$.


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


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