×

Measurable sequences. (English) Zbl 1452.11013

In the paper numerous interrelations between several notions connected with the distribution of sequences in various structures (real numbers, integers, ring of polyadic numbers) are described, in particular between various variants of the notion of a distribution function. For instance, between asymptotic distribution function from the theory of uniform distribution of sequences and distribution functions of random variables, or connection between independence in the sense of probability theory and statistical independence of sequences in case of continuous distribution functions, etc. The techniques used in the paper employ tools from the theory of the uniform distribution in \(\mathbb{R}\) (H. Weyl) or in \(\mathbb{Z}\) (I. Niven), that from the theory of Buck measure density theory, the asymptotic density, from probabilistic number theory, or the topology of polyadic numbers.

MSC:

11B05 Density, gaps, topology
11K06 General theory of distribution modulo \(1\)
11K31 Special sequences
11J71 Distribution modulo one
11B25 Arithmetic progressions