Monatshefte für Mathematik

, Volume 160, Issue 1, pp 1–29

Irregular discrepancy behavior of lacunary series

Article

Abstract

In 1975 Philipp showed that for any increasing sequence (nk) of positive integers satisfying the Hadamard gap condition nk+1/nk > q > 1, k ≥ 1, the discrepancy DN of (nkx) mod 1 satisfies the law of the iterated logarithm
$$ 1/4 \leq {\mathop {\rm lim\,sup} \limits _{N\to\infty}}\, N D_N(n_k x) (N \log \log N)^{-1/2}\leq C_q\quad \textup{a.e.}$$
Recently, Fukuyama computed the value of the lim sup for sequences of the form nk = θk, θ > 1, and in a preceding paper the author gave a Diophantine condition on (nk) for the value of the limsup to be equal to 1/2, the value obtained in the case of i.i.d. sequences. In this paper we utilize this number-theoretic connection to construct a lacunary sequence (nk) for which the lim sup in the LIL for the star-discrepancy \({D_N^*}\) is not a constant a.e. and is not equal to the lim sup in the LIL for the discrepancy DN.

Keywords

Discrepancy Lacunary series Law of the iterated logarithm 

Mathematics Subject Classification (2000)

Primary: 11K38 42A55 60F15 

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Copyright information

© Springer-Verlag 2008

Authors and Affiliations

  1. 1.Institute of Mathematics AGraz University of TechnologyGrazAustria

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