Irregular discrepancy behavior of lacunary series II

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 root 2) <= lim sup(N ->infinity) ND(N)(n(k)x)(2N log log N)(-1/2) <= C(q) for almost all x is an element of (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.