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On the law of the iterated logarithm for the discrepancy of sequences $\langle n_{\mathbf{k}} x \rangle$ with multidimensional indices

Author(s): Christoph Aistleitner

Journal: Uniform Distribution Theory
ISSN 1336-913X

Volume: 2;
Issue: 2;
Start page: 89;
Date: 2007;
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Keywords: Discrepancy | lacunary series | law of the iterated logarithm

By a classical result of Weyl (1916), for any increasingsequence $(n_k)$ of positive integers, $(n_kx)$ is uniformlydistributed mod 1 for almost all $x$. The precise asymptotics ofthe discrepancy of this sequence is known only in a few cases,e.g., for $n_k=k$ (Khintchine (1924)) and for lacunary $(n_k)$(Philipp (1975)). In this paper we extend Philipp's result tolacunary sequences with multidimensional indices.
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