Extremal discrepancy behavior of lacunary sequences

Abstract

In 1975 Walter Philipp proved the law of the iterated logarithm (LIL) for the discrepancy of lacunary sequences: for any sequence $(n_k)_{k \ge 1}$ satisfying the Hadamard gap condition $n_{k+1} / n_k \ge q > 1,~k \ge 1,$ we have

$$\begin{aligned} \frac{1}{4 \sqrt{2}} \le \limsup _{N \rightarrow \infty } \frac{N D_N(\{ n_1 x \}, \dots , \{n_N x\})}{\sqrt{2 N \log \log N}} \le C_q \end{aligned}$$

for almost all $x$. In recent years there has been significant progress concerning the precise value of the limsup in this LIL for special sequences $(n_k)_{k \ge 1}$ having a “simple” number-theoretic structure. However, since the publication of Philipp’s paper there has been no progress concerning the lower bound in this LIL for generic lacunary sequences $(n_k)_{k \ge 1}$. The purpose of the present paper is to collect known results concerning this problem, to investigate what the optimal value in the lower bound could be, and for which special sequences $(n_k)_{k \ge 1}$ a small value of the limsup in this LIL can be obtained. We formulate three open problems, which could serve as the main targets for future research.