Publications Meetings The Profession Membership Programs Math Samplings Policy & Advocacy In the News About the AMS
   
Mobile Device Pairing
Green Open Access
Transactions of the American Mathematical Society
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
Full-text PDF

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 $ 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.


References [Enhancements On Off] (What's this?)


Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC (2010): 11K38, 60F15, 11D04, 11J83, 42A55

Retrieve articles in all journals with MSC (2010): 11K38, 60F15, 11D04, 11J83, 42A55


Additional Information

Christoph Aistleitner
Affiliation: Institute of Mathematics A, Graz University of Technology, Steyrergasse 30, 8010 Graz, Austria
Email: aistleitner@math.tugraz.at

DOI: http://dx.doi.org/10.1090/S0002-9947-2012-05740-0
PII: S 0002-9947(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.