Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



On the law of the iterated logarithm for the discrepancy of lacunary sequences II

Author: Christoph Aistleitner
Journal: Trans. Amer. Math. Soc. 365 (2013), 3713-3728
MSC (2010): Primary 11K38, 60F15, 11D04, 11J83, 42A55
Published electronically: October 31, 2012
MathSciNet review: 3042600
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Abstract: By a classical heuristics, lacunary function systems exhibit many asymptotic properties which are typical for systems of independent random variables. For example, for a large class of functions $ f$ the system $ (f(n_k x))_{k \geq 1}$, where $ (n_k)_{k \geq 1}$ is a lacunary sequence of integers, satisfies a law of the iterated logarithm (LIL) of the form

$\displaystyle c_1 \leq \limsup _{N \to \infty } \frac {\sum _{k=1}^N f(n_k x)}{\sqrt {2 N \log \log N}} \leq c_2 \qquad \textup {a.e.},$ (1)

where $ c_1,c_2$ are appropriate positive constants. In a previous paper we gave a criterion, formulated in terms of the number of solutions of certain linear Diophantine equations, which guarantees that the value of the $ \limsup $ in (1) equals the $ L^2$-norm of $ f$ for $ \textup {a.e.}$ $ x$, which is exactly what one would also expect in the case of i.i.d. random variables. This result can be used to prove a precise LIL for the discrepancy of $ (n_k x)_{k \geq 1}$, which corresponds to the Chung-Smirnov LIL for the Kolmogorov-Smirnov-statistic of i.i.d. random variables.

In the present paper we give a full solution of the problem in the case of ``stationary'' Diophantine behavior, by this means providing a unifying explanation of the aforementioned ``regular'' LIL behavior and the ``irregular'' LIL behavior which has been observed by Kac, Erdős, Fortet and others.

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Additional Information

Christoph Aistleitner
Affiliation: Institute of Mathematics A, Graz University of Technology, Steyrergasse 30, 8010 Graz, Austria

Keywords: Discrepancy, law of the iterated logarithm, Diophantine equations, lacunary series
Received by editor(s): July 21, 2011
Received by editor(s) in revised form: November 1, 2011
Published electronically: October 31, 2012
Additional Notes: The author’s research was supported by the Austrian Research Foundation (FWF), Project S9603-N23.
Article copyright: © Copyright 2012 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.