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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

On permutations of Hardy-Littlewood-Pólya sequences


Authors: Christoph Aistleitner, István Berkes and Robert F. Tichy
Journal: Trans. Amer. Math. Soc. 363 (2011), 6219-6244
MSC (2000): Primary 42A55, 11K60, 60F05, 60F15
Published electronically: July 22, 2011
MathSciNet review: 2833551
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Abstract: Let $ {\mathcal H}=(q_1, \ldots, q_r)$ be a finite set of coprime integers and let $ n_1, n_2, \ldots$ denote the multiplicative semigroup generated by $ \mathcal H$ and arranged in increasing order. The distribution of such sequences has been studied intensively in number theory, and they have remarkable probabilistic and ergodic properties. In particular, the asymptotic properties of the sequence $ \{n_kx\}$ are similar to those of independent, identically distributed random variables; here $ \{\cdot \}$ denotes fractional part. In this paper we prove that under mild assumptions on the periodic function $ f$, the sequence $ f(n_kx)$ obeys the central limit theorem and the law of the iterated logarithm after any permutation of its terms. Note that the permutational invariance of the CLT and LIL generally fails for lacunary sequences $ f(m_kx)$ even if $ (m_k)$ has Hadamard gaps. Our proof depends on recent deep results of Amoroso and Viada on Diophantine equations. We will also show that $ \{n_kx\}$ satisfies a strong independence property (``interlaced mixing''), enabling one to determine the precise asymptotic behavior of permuted sums $ S_N (\sigma)= \sum_{k=1}^N f(n_{\sigma(k)} x)$.


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Additional Information

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

István Berkes
Affiliation: Institute of Statistics, Graz University of Technology, Münzgrabenstrasse 11, A–8010 Graz, Austria
Email: berkes@tugraz.at

Robert F. Tichy
Affiliation: Department of Mathematics A, Graz University of Technology, Steyrergasse 30, A–8010 Graz, Austria
Email: tichy@tugraz.at

DOI: http://dx.doi.org/10.1090/S0002-9947-2011-05490-5
PII: S 0002-9947(2011)05490-5
Keywords: Lacunary series, mixing, central limit theorem, law of the iterated logarithm, Diophantine equations.
Received by editor(s): September 10, 2009
Published electronically: July 22, 2011
Additional Notes: The first author’s research was supported by Austrian Science Fund Grant No. S9603-N23 and an MOEL scholarship of the Österreichisch Forschungsgemeinschaft
The second author’s research was supported by Austrian Science Fund Grant No. S9603-N23 and OTKA grants K 67961 and K 81928
The third author’s research was supported by Austrian Science Fund Grant No. S9603-N23
Dedicated: Dedicated to the memory of Walter Philipp
Article copyright: © Copyright 2011 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.