Irregular discrepancy behavior of lacunary series

In 1975 Philipp showed that for any increasing sequence (n(k)) of positive integers satisfying the Hadamard gap condition n(k+1)/n(k) > q > 1, k >= 1, the discrepancy D(N) of (n(k)x) mod 1 satisfies the law of the iterated logarithm 1/4 <= N -> infinity lim sup N ND(N) (n(k)x)(N log log N)(-1/2) <= C(q) a.e. Recently, Fukuyama computed the value of the lim sup for sequences of the form n(k) = theta(k), theta > 1, and in a preceding paper the author gave a Diophantine condition on (n(k)) 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 (n(k)) 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 D(N).