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A Note on a Problem of Motzkin Regarding
Density of Integral Sets with Missing
Differences Ram Krishna Pandey¹
Department of Mathematics
Indian Institute of Technology
Patliputra Colony, Patna - 800013
India
mailto:ram@iitp.ac.inram@iitp.ac.in

Amitabha Tripathi²
Department of Mathematics
Indian Institute of Technology
Hauz Khas, New Delhi - 110016
India
mailto:atripath@maths.iitd.ac.inatripath@maths.iitd.ac.in

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