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Remarks on uniform density of sets of integers. (English) Zbl 1012.11012

Let \(A\subseteq\mathbb{N}=\{1,2,3,\dots\}\) and \(m,n,\in\mathbb{N}\), \(m<n\). Denote by \(A(m,n)\) the cardinality of \(A\cap [m,n]\) and define \(\alpha_s=\min_{t\geq 0}A(t+1,t+s)\) and \(\alpha^s=\max_{t\geq 0}A(t+1,t+s)\). Then there exist \(\underline{u}(A)=\lim_{s\to\infty}\frac{\alpha_s}s\), \(\overline{u}(A)=\lim_{s\to\infty}\frac{\alpha^s}s\), and they are called the lower and the upper uniform density of \(A\), resp. In the case \(\underline{u}(A)=\overline{u}(A)\) we have the uniform density \(u(A)\). In this paper some properties of this concept are studied, and it is shown that the uniform density has the Darboux property. This is defined by the following: Let \(v\) be a nonnegative set function defined on a class \(S\leq 2^N\). The function \(v\) is said to have the Darboux property provided that, if \(v(A)>0\) for \(A\in S\) and \(0<t<v(A)\), then there is a set \(B\subseteq A\), \(B\in S\) such that \(v(B)=t\).

MSC:

11B05 Density, gaps, topology
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