Irregular LIL behaviour of lacunary trigonometric series (Q1176048)

From the work of \textit{P. Erdős} [Publ. Math. Inst. Hung. Acad. Sci., Ser. A 7, 37--42 (1962; Zbl 0116.04703)] we know that, for slowly increasing \(n_ k\), the behavior of \(\cos 2\pi n_k x\) becomes pathological. The purpose of this paper is to present a further surprising example. The author constructs a sequence \((n_k)\) of integers satisfying the gap condition \(n_{k+1}/n_k\geq 1+ c_k/\sqrt k\) with \(c_k\to\infty\) such that \(\cos 2\pi n_k x\) obeys both the central limit theorem and the ordinary law of the iterated logarithm, but fails to obey the Kolmogorov upper-lower limit test, and exhibits asymmetric fluctuation properties: for any \(\lambda>0\) and almost all \(x\) we have \[ \sum_{k\leq N} \cos 2\pi n_k x < \sqrt N (\log\log N-\lambda\log_3 N)^{1/2}\quad\text{for }N\geq N_0 \] \[ \sum_{k\leq N}(-\cos 2\pi n_k x) \geq \sqrt N (\log\log N+\lambda\log_3 N)^{1/2} \] for infinitely many \(N\). Here \(\log_3\) denotes the three times iterated logarithm.