The Khinchin inequality and Chen’s theorem

Skriganov, M.

doi:10.1090/S1061-0022-2012-01216-1

The Khinchin inequality and Chen’s theorem
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by M. M. Skriganov
Translated by: the author

St. Petersburg Math. J. 23 (2012), 761-778

DOI: https://doi.org/10.1090/S1061-0022-2012-01216-1

Published electronically: April 13, 2012

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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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