Monatshefte für Mathematik

, Volume 160, Issue 1, pp 1–29

Irregular discrepancy behavior of lacunary series

Authors

    • Institute of Mathematics AGraz University of Technology
Article

DOI: 10.1007/s00605-008-0067-x

Cite this article as:
Aistleitner, C. Monatsh Math (2010) 160: 1. doi:10.1007/s00605-008-0067-x

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

DiscrepancyLacunary seriesLaw of the iterated logarithm

Mathematics Subject Classification (2000)

Primary: 11K3842A5560F15

Copyright information

© Springer-Verlag 2008