Irregular discrepancy behavior of lacunary series

In 1975 Philipp showed that for any increasing sequence (nk) of positive integers satisfying the Hadamard gap condition nk+1/nk > q > 1, k ≥ 1, the discrepancy DN of (nkx) mod 1 satisfies the law of the iterated logarithm 1/4 ≤ lim sup N→∞ NDN(nkx)(N log logN) −1/2 ≤ Cq a.e. Recently, Fukuyama computed the value of the lim sup for sequences of the form nk = θ , θ > 1, and in a preceding paper the author gave a Diophantine condition on (nk) 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 (nk) 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 DN .