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

Author: Christoph Aistleitner
Journal: Trans. Amer. Math. Soc. 362 (2010), 5967-5982
MSC (2000): Primary 11K38, 42A55, 60F15
Published electronically: June 16, 2010
MathSciNet review: 2661504
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Abstract: A classical result of Philipp (1975) states that for any sequence $ (n_k)_{k \geq 1}$ of integers satisfying the Hadamard gap condition $ n_{k+1}/n_k\ge q>1 (k=1, 2, \ldots)$, the discrepancy $ D_N$ of the sequence $ (n_k x)_{k\ge 1}$ mod 1 satisfies the law of the iterated logarithm (LIL), i.e.

$\displaystyle 1/4 \leq \limsup_{N \to \infty} N D_N(n_k x) (N \log \log N)^{-1/2} \leq C_q \quad {a.e.}$

The value of the $ \limsup$ is a long-standing open problem. Recently Fukuyama explicitly calculated the value of the $ \limsup$ for $ n_k= \theta^k$, $ \theta>1$, not necessarily integer. We extend Fukuyama's result to a large class of integer sequences $ (n_k)$ characterized in terms of the number of solutions of a certain class of Diophantine equations and show that the value of the $ \limsup$ is the same as in the Chung-Smirnov LIL for i.i.d. random variables.

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Additional Information

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

Keywords: Discrepancy, lacunary series, law of the iterated logarithm
Received by editor(s): June 2, 2008
Received by editor(s) in revised form: February 19, 2009
Published electronically: June 16, 2010
Additional Notes: This research was supported by the Austrian Research Foundation (FWF), Project S9603-N13.
Article copyright: © Copyright 2010 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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