In 1975 Philipp proved the following law of the iterated logarithm (LIL) for the discrepancy of lacunary series: let (n_{k})_{k ≥ 1} be a lacunary sequence of positive integers, i.e. a sequence satisfying the Hadamard gap condition n_{k+1}/n_{k} > 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” D_{N} 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.

Keywords

DiscrepancyLacunary seriesLaw of the iterated logarithm