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
DOI:
https://doi.org/10.1090/S0002-9947-2012-05740-0
Published electronically:
October 31, 2012
MathSciNet review:
3042600
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Abstract | References | Similar Articles | Additional Information
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 the system
, where
is a lacunary sequence of integers, satisfies a law of the iterated logarithm (LIL) of the form
![]() | (1) |
where







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
Email:
aistleitner@math.tugraz.at
DOI:
https://doi.org/10.1090/S0002-9947-2012-05740-0
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.