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

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 of positive integers, a problem of Motzkin asks to
determine the maximal density among sets of nonnegative
integers in which no two elements differ by an element of . The
problem is completely settled when , and some partial
results are known for several families of when . In 1985
Rabinowitz & Proulx provided a lower bound for
and conjectured that their bound was sharp. Liu & Zhu proved this
conjecture in 2004. For each , we determine
, which is a lower bound for
, and conjecture this to be the exact value of
.

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

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