, Volume 160, Issue 1, pp 1-29
Date: 07 Oct 2008

Irregular discrepancy behavior of lacunary series

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Abstract

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

$$ 1/4 \leq {\mathop {\rm lim\,sup} \limits _{N\to\infty}}\, N D_N(n_k x) (N \log \log N)^{-1/2}\leq C_q\quad \textup{a.e.}$$
Recently, Fukuyama computed the value of the lim sup for sequences of the form n k = θ k , θ > 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 .