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