## On the density of integral sets with missing differences from sets related to arithmetic progressions.(English)Zbl 1229.11044

Given a set $$M$$ of positive integer, a set $$S$$ of non-negative integers is said to be an $$M$$-set if $$a\in S$$ and $$b\in S$$ imply $$a-b\notin M$$. In an unpublished problem collection T. S. Motzkin asks for determining the quantity $$\mu(M)=\bar{\delta}(S)$$ where $$S$$ varies over all $$M$$-sets and $$\bar{\delta}(S)$$ stands for the upper asymptotic density. In the case $$|M|\leq2$$ the problem was completely solved by D. G. Cantor and B. Gordon [J. Comb. Theory, Ser. A 14, 281–287 (1973; Zbl 0277.10043)], and there are known partial answers for several families of $$M$$ with $$|M|\geq 3$$. In the present paper some cases when $$M$$ either contains an arithmetic progression or is contained in an arithmetic progression are settled.

### MSC:

 11B83 Special sequences and polynomials 11B05 Density, gaps, topology 11B25 Arithmetic progressions

Zbl 0277.10043
Full Text:

### References:

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