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Maximal upper asymptotic density of sets of integers with missing differences from a given set. (English) Zbl 1349.11013

Summary: Let \(M\) be a given nonempty set of positive integers and \(S\) any set of nonnegative integers. Let \(\overline \delta (S)\) denote the upper asymptotic density of \(S\). We consider the problem of finding \[ \mu (M):=\sup_{S}\overline\delta (S), \] where the supremum is taken over all sets \(S\) satisfying that for each \(a,b\in S\), \(a-b\notin M\). In this paper we discuss the values and bounds of \(\mu (M)\) where \(M=\{a,b,a+nb\}\) for all even integers and for all sufficiently large odd integers \(n\) with \(a<b\) and \(\gcd (a,b)=1\).

MSC:

11B05 Density, gaps, topology
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