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A note on uniform or Banach density. (English) Zbl 1239.11012

Let \({\mathbb N}\) be the set of positive integers, \(A\subset{\mathbb N}\) and \(I=[s,t]\subset{\mathbb N}\) an interval of length \(|I|=t-s\) and \(A(s,t)=A\cap[s,t]\). As a measure of a subset \(A\subset{\mathbb N}\) various density concepts are used, among them the Banach and the uniform density. The notions of the upper Banach density is defined in the paper as \[ \overline{b}(A)=\sup\{x\in[0,1];\;\forall\ell\in{\mathbb N}\;\exists I\subset{\mathbb N}: |I|\geq \ell \wedge |A\cap I|/|I|\geq x\}. \] The notion of the upper uniform density can be found in the literature in two forms, either as \[ \overline{a}(A)=\lim_{s\to\infty}(\limsup_{n\to\infty} A(n+1,n+s))/s \] or as \[ \overline{c}(A)=\lim_{s\to\infty}(\sup_{n\to\infty} A(n+1,n+s))/s. \] The aim of the paper is to show that all three values coincide. Dual results for the lower densities are also proved.

MSC:

11B05 Density, gaps, topology
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