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The Khinchin inequality and Chen's theorem


Author: M. M. Skriganov
Translated by: the author
Original publication: Algebra i Analiz, tom 23 (2011), nomer 4.
Journal: St. Petersburg Math. J. 23 (2012), 761-778
MSC (2010): Primary 11K36
DOI: https://doi.org/10.1090/S1061-0022-2012-01216-1
Published electronically: April 13, 2012
MathSciNet review: 2893524
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Abstract: Chen's theorem on the mean values of $ L_q$-discrepancies is one of the basic results in the theory of uniformly distributed point sets. This is a difficult result, based on deep and nontrivial combinatorial arguments (see the papers by Chen and Beck on irregularities of distributions). The paper is aimed at showing that the results of such a type are intimately related to lacunarity and statistical independence of certain function series. In particular, the classical Khinchin inequality for the Rademacher functions is employed to prove an important generalization of Chen's theorem. In a forthcoming paper, the author will continue the study of the phenomena of lacunarity and statistical independence in the context of the theory of uniformly distributed point sets.


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Additional Information

M. M. Skriganov
Affiliation: St. Petersburg Branch, Steklov Mathematical Institute, Russian Academy of Sciences, Fontanka 27, St. Petersburg 191023, Russia
Email: skrig@pdmi.ras.ru

DOI: https://doi.org/10.1090/S1061-0022-2012-01216-1
Keywords: Uniform distributions, harmonic analysis, lacunary series
Received by editor(s): January 27, 2011
Published electronically: April 13, 2012
Additional Notes: Partially supported by RFBR (project no. 08-01-00182)
Dedicated: To Vasiliĭ Mikhaĭlovich Babich, on the occasion of his 80th birthday, with unwavering admiration
Article copyright: © Copyright 2012 American Mathematical Society