Monatshefte für Mathematik

, Volume 161, Issue 3, pp 255–270

Irregular discrepancy behavior of lacunary series II


DOI: 10.1007/s00605-009-0165-4

Cite this article as:
Aistleitner, C. Monatsh Math (2010) 161: 255. doi:10.1007/s00605-009-0165-4


In 1975 Philipp proved the following law of the iterated logarithm (LIL) for the discrepancy of lacunary series: let (nk)k ≥ 1 be a lacunary sequence of positive integers, i.e. a sequence satisfying the Hadamard gap condition nk+1/nk > q > 1. Then \({1/(4 \sqrt{2}) \leq \limsup_{N \to \infty} N D_N(n_k x) (2 N \log \log N)^{-1/2}\leq C_q}\) for almost all \({x \in (0,1)}\) in the sense of Lebesgue measure. The same result holds, if the “extremal discrepancy” DN is replaced by the “star discrepancy” \({D_N^*}\) . It has been a long standing open problem whether the value of the limsup in the LIL has to be a constant almost everywhere or not. In a preceding paper we constructed a lacunary sequence of integers, for which the value of the limsup in the LIL for the star discrepancy is not a constant a.e. Now, using a refined version of our methods from this preceding paper, we finally construct a sequence for which also the value of the limsup in the LIL for the extremal discrepancy is not a constant a.e.


DiscrepancyLacunary seriesLaw of the iterated logarithm

Mathematics Subject Classification (2000)

Primary 11K3842A5560F15

Copyright information

© Springer-Verlag 2009

Authors and Affiliations

  1. 1.Institute of Mathematics AGraz University of TechnologyGrazAustria