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On the density of integral sets with missing differences. (English) Zbl 1218.11009

Landman, Bruce (ed.) et al., Combinatorial number theory. Proceedings of the 3rd ‘Integers Conference 2007’, Carrollton, GA, USA, October 24–27, 2007. Berlin: Walter de Gruyter (ISBN 978-3-11-020221-2/hbk), Integers 9, Suppl., Article A12, 157-169 (2009).
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 In an unpublished problem collection T. S. Motzkin asked for determining the quantity \(\mu(M)=\bar{\delta}(S)\), where \(S\) varies over all \(M\)-sets and \(\bar{\delta}(S)\)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.
This paper contains a further contribution of the authors [cf. e.g. J. Integer Seq. 14, No. 6, Article 11.6.3, 8 p., electronic only (2011; Zbl 1239.11013) or J. Number Theory 131, No. 4, 634–647 (2011; Zbl 1229.11044)] to this open problem, where they consider the case when \(M=\{a,b,c\}\), with \(c\) a multiple of \(a\) or \(b\). In most cases they prove lower bounds for \(\mu(M)\), which they conjecture to coincide with the exact values, while in some case they give the exact values.
For the entire collection see [Zbl 1159.11002].

MSC:

11B05 Density, gaps, topology
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