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Uniform distribution of sequences with respect to the Voronoi and Riesz methods. (English. Russian original) Zbl 1136.11048
Dokl. Math. 74, No. 2, 635-636 (2006); translation from Dokl. Akad. Nauk., Ross. Akad. Nauk. 410, No. 1, 19-20 (2006).
In the paper under review the author considers some of the problems proposed by V. V. Kozlov and T. Madsen [Sb. Math. 196, No. 10, 1495–1502 (2005); translation from Mat. Sb. 196, No. 10, 103–110 (2005; Zbl 1142.11342)]. The main results are as follows.
(i) If a sequence $$x_n\in [0,1]$$ is uniformly distributed with respect to some regular Voronoi method $$(W,q_n)$$, then it is uniformly distributed in the sense of Weyl, i.e., with respect to the Cesàro method $$(C,1)$$.
(ii) If $$\theta$$ is irrational, then the sequence of fractional parts $$\{n\theta\}$$ $$(n= 1,2,\dots)$$ is uniformly distributed with respect to the Voronoi method $$(W,q_n)$$, if and only if it is uniformly distributed with respect to the Riesz method $$(R,q_n)$$. For example, if the sequence $$q_n$$ $$(n= 1,2,\dots)$$ is monotone, then the sequence $$\{n\theta\}$$ $$(n= 1,2,\dots)$$ is uniformly distributed with respect to both methods $$(W,q_n)$$ and $$(R,q_n)$$ for any irrational number $$\theta$$.

##### MSC:
 11K06 General theory of distribution modulo $$1$$ 37A45 Relations of ergodic theory with number theory and harmonic analysis (MSC2010) 40G99 Special methods of summability
Zbl 1142.11342
Full Text:
##### References:
 [1] V. V. Kozlov and T. Madsen, Mat. Sb. 196(10), 103–110 (2005). [2] G. H. Hardy, Divergent Series, 2nd ed. (Clarendon, Oxford, 1956; Inostrannaya Literatura, Moscow, 1951). · Zbl 0897.01044 [3] L. Kuipers and G. Niderreiter, Uniform Distribution of Sequences (Wiley, New York, 1974; Nauka, Moscow, 1985). [4] M. Tsuji, J. Math. Soc. Japan 4, 313–322 (1952). · Zbl 0048.03302 [5] A. P. Prudnikov, Yu. A. Brychkov, and O. I. Marichev, Integrals and Series (Nauka, Moscow, 1981) [in Russian]. · Zbl 0511.00044
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