A metric discrepancy result for lacunary sequences (Q2880635)

In 1975, [Acta Arith. 26, 241--251 (1975; Zbl 0263.10020)] \textit{W. Philipp} proved the bounded law of the iterated logarithm (LIL) for the discrepancy of lacunary sequences. In 2008, the first author of the present paper under review developed a new technique to calculate the exact value of the limsup in the LIL for special lacunary sequences; it turns out that the a.e. value of this limsup depends on number-theoretic properties of the specific lacunary sequence in a very sensitive and interesting way.NEWLINENEWLINEIn the present paper under review, the authors prove that any constant \(\sigma \geq 1/2\) can be the a.e. limsup in the LIL for an appropriate lacunary sequence. For the proof they use an approximation of so-called Hardy--Littlewood--Polya sequences (which are sequences generated by a finite set of primes; it is well-known that they exhibit probabilistic properties very similar to those of lacunary sequences, see e.g. [\textit{W. Philipp}, Trans. Am. Math. Soc. 345, No. 2, 705--727 (1994; Zbl 0812.11045)]) together with a randomization method.