Irregular discrepancy behavior of lacunary series II

Abstract

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.