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 CesĂ ro 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 \(\theta\).


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: DOI


[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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