Metric discrepancy results for geometric progressions and variations (Q2917751)

The first part of this paper is a survey on classical and recent results concerning the law of the iterated logarithm for the discrepancy of lacunary series. In the second part, it is shown that the limsup in the law of the iterated logarithm for fractional parts of geometric progressions \(\{ \theta^k x \}\) is constant almost everywhere, for \(|\theta|>1\). This was already shown by the same author in an earlier paper for a large class of values \(\theta>1\), together with a calculation of the precise value of the limsup see [Acta Math. Hung. 118, No. 1--2, 155--170 (2008; Zbl 1241.11090)]. The precise values of the limsups in the case \(\theta<-1\) are announced in the present paper for a large class of values of \(\theta\), with proofs given elsewhere.