Weighted averages, uniform distribution, and strict ergodicity. (English) Zbl 1145.11057

Kozlov, V. V.

Russ. Math. Surv. 60, No. 6, 1121-1146 (2005); translation from Usp. Mat. Nauk 60, No. 6, 115-138 (2005).

The article gives an overview of various generalizations of the classical concept of uniform distribution. The first digit paradox (Benford’s law) is discussed and uniform distribution with respect to Riesz and Voronoi summation methods is considered. Using these generalizations of Cesàro summability ergodic theorems and variants of the strong law of large numbers and the iterated logarithm are given. The last chapter lists some open problems.

Reviewer: Martin Blümlinger (Wien)

MSC:

11K36	Well-distributed sequences and other variations
28D05	Measure-preserving transformations
60F15	Strong limit theorems

Keywords:

uniform distribution; Riesz convergence; Voronoi convergence; strict ergodicity; weighted averages; Strong law of large numbers; ergodic theorem; Benford’s law