## 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

### Keywords:

Banach density; uniform density
Full Text:

### References:

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