Bounded law of the iterated logarithm for discrepancies of permutations of lacunary sequences (Q2917753)

In 1975, \textit{W. Philipp} [Acta Arith. 26, 241--251 (1975; Zbl 0263.10020)] proved that for a so-called lacunary sequence \((n_k)_{k \geq 1}\) satisfying the gap condition NEWLINE\[NEWLINE \frac{n_{k+1}}{n_k} \geq q > 1, \qquad k \geq 1, NEWLINE\]NEWLINE the law of the iterated logarithm (LIL) for the discrepancy holds in the form NEWLINE\[NEWLINE \frac{1}{4 \sqrt{2}} \leq \limsup_{N \to \infty} \frac{N D_N(\{n_1 x\}, \dots, \{n_N x\})}{\sqrt{2 N \log \log N}} \leq \frac{1}{\sqrt{2}} \left( 166 + \frac{644}{\sqrt{q}-1} \right) \quad \text{a.e.} NEWLINE\]NEWLINE The reviewer [Trans. Am. Math. Soc. 362, No. 11, 5967-5982 (2010; Zbl 1209.11072)] improved the value on the right-hand side of this LIL to NEWLINE\[NEWLINE \frac{1}{2} + \frac{6}{q^{1/4}}. NEWLINE\]NEWLINE In the present paper, this is further improved to NEWLINE\[NEWLINE \left( \frac{1}{4} + \frac{1}{\sqrt{3}(q-1)}\right)^{1/2}. NEWLINE\]NEWLINE Comparing this with the known exact value of the \(\limsup\) in the special case \(n_k=q^k,~k \geq 1\), it is seen that this upper bound gives the optimal speed of convergence of the limsup to 1/2 as \(q \to \infty\), up to a multiplicative constant. Furthermore, the authors show that their results also hold if the terms of the sequence \((n_k)_{k \geq 1}\) are permuted.