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\vskip 1cm{\LARGE\bf 
A Note on a Problem of Motzkin Regarding \\
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Density of Integral Sets with Missing \\
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Differences 
}
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\large
Ram Krishna Pandey\thanks{This work was done when at Department of Mathematics, Indian Institute of Technology, Delhi} \\
Department of Mathematics \\
Indian Institute of Technology \\
Patliputra Colony, Patna -- 800013 \\
India\\
\href{mailto:ram@iitp.ac.in}{\tt ram@iitp.ac.in}\\
\ \\
Amitabha Tripathi\thanks{\em Corresponding author} \\
Department of Mathematics \\
Indian Institute of Technology \\
Hauz Khas, New Delhi -- 110016 \\
India \\
\href{mailto:atripath@maths.iitd.ac.in}{\tt atripath@maths.iitd.ac.in}
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\begin{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 $|M| \le 2$, and some partial
results are known for several families of $M$ when $|M| \ge 3$. 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
${\k}(\{a,b,n(a+b)\})$, which is a lower bound for
$\mu(\{a,b,n(a+b)\})$, and conjecture this to be the exact value of
${\mu}(\{a,b,n(a+b)\})$.
\end{abstract}


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