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On the Uniform Theory of Lacunary Series

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Number Theory – Diophantine Problems, Uniform Distribution and Applications

Abstract

The theory of lacunary series starts with Weierstrass’ famous example (1872) of a continuous, nondifferentiable function and now we have a wide and nearly complete theory of lacunary subsequences of classical orthogonal systems, as well as asymptotic results for thin subsequences of general function systems. However, many applications of lacunary series in harmonic analysis, orthogonal function theory, Banach space theory, etc. require uniform limit theorems for such series, i.e., theorems holding simultaneously for a class of lacunary series, and such results are much harder to prove than dealing with individual series. The purpose of this paper is to give a survey of uniformity theory of lacunary series and discuss new results in the field. In particular, we study the permutation-invariance of lacunary series and their connection with Diophantine equations, uniform limit theorems in Banach space theory, resonance phenomena for lacunary series, lacunary sequences with random gaps, and the metric discrepancy theory of lacunary sequences.

Dedicated to Professor Robert Tichy on the occasion of his 60th birthday

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Notes

  1. 1.

    This is meant as \(\lim _{n\rightarrow \infty }\mathbb{E}(X_{n}Y ) = 0\) for all Y ∈ L q where 1∕p + 1∕q = 1. This convergence should not be confused with weak convergence of probability distributions, also called convergence in distribution.

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Acknowledgement

The research is supported by FWF grant P24302-N18 and NKFIH grant K 108615.

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Authors and Affiliations

  1. Graz University of Technology, Institute of Statistics, Kopernikusgasse 24, 8010, Graz, Austria

    István Berkes

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  1. István Berkes
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Correspondence to István Berkes .

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Editors and Affiliations

  1. Institute of Analysis and Number Theory, Graz University of Technology, Graz, Austria

    Christian Elsholtz

  2. Institute of Analysis and Number Theory, Graz University of Technology, Graz, Austria

    Peter Grabner

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Berkes, I. (2017). On the Uniform Theory of Lacunary Series. In: Elsholtz, C., Grabner, P. (eds) Number Theory – Diophantine Problems, Uniform Distribution and Applications. Springer, Cham. https://doi.org/10.1007/978-3-319-55357-3_6

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