## Covering densities.(English)Zbl 0770.11009

Let $$D_ 0$$ be the algebra of sets generated by all infinite arithmetic progressions $$\{a+bd\mid d=1,2,\dots;\;a,b\in\mathbb{N}_ 0\}$$. R. C. Buck’s measure density is defined by considering natural density as a finitely additive measure on $$D_ 0$$ and extending it to a finitely additive outer probability measure $$\mu^*$$, defined for all sets $$S\subseteq\mathbb{N}$$, which determines a field $$D_ \mu$$ of $$\mu^*$$- measurable sets.
The author generalizes this theory by restricting the moduli $$d$$ to a fixed (infinite) set $$A\subseteq\mathbb{N}$$ of values, assumed to contain all divisors and all l.c.m’s of its elements (and pairs), studying the field $$D_ A$$ and the outer measure $$\mu^*_ A$$ so obtained. Typical results:
(1) $$\forall S\in D_ A$$ $$\forall\alpha\in[0,\mu^*_ A(S)]$$ $$\exists S_ 1\subseteq S:S_ 1\in D_ A$$, $$\mu^*_ A(S_ 1)=\alpha$$ (Darboux property).
(2) For any $$S\subseteq\mathbb{N}$$, $$\mu^*(S)=1$$ iff a suitable re- arrangement of $$S$$ is uniformly distributed to all moduli $$d\in A$$: ($$A$$- u.d.).
(3) A set $$S\subseteq\mathbb{N}$$ is $$\mu^*_ A$$-measurable if it intersects with each $$A$$-u.d. sequence in a set of natural density $$\mu^*_ A(S)$$.

### MSC:

 11B05 Density, gaps, topology 11B25 Arithmetic progressions 11B50 Sequences (mod $$m$$)
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### References:

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