Irregular discrepancy behavior of lacunary series. II (Q711630)

Answering a problem of P. Erdős, \textit{W. Philipp} [Acta Arith. 26, 241--251 (1975; Zbl 0263.10020)] proved the following law of the iterated logarithm (LIL) for the discrepancy of lacunary series: let \((n_{k})_{k\geq 1}\) be a lacunary sequence of positive integers, i.e. a sequence satisfying the Hadamard gap condition \(n_{k+1}/n_{k}> q > 1,\) then \[ 1/(4\sqrt{2})\leq \limsup_{N\rightarrow\infty}ND_{N}(n_{k}x)(2N\log\log N)^{-1/2}\leq C_{q} \] for almost all \(x\in (0,1)\) in the sense of Lebesgue measure. The same result holds, if the ``extremal discrepancy'' \(D_{N}\) is replaced by the ``star discrepancy'' \(D_{N}^{*}.\) It has been an open problem whether the value of the \(\limsup\) in the LIL has to be a constant almost everywhere or not. In a preceding paper [Monatsh. Math. 160, No. 1, 1--29 (2010; Zbl 1197.11089)], C. Aistleitner constructed a lacunary sequence of integers, for which the value of the \(\limsup\) in the LIL for the star discrepancy is not a constant a.e. In the present paper the author constructs a sequence for which also the value of the \(\limsup\) in the LIL for the extremal discrepancy is not a constant a.e.