Irregular discrepancy behavior of lacunary series
In 1975 \textit{W. Philipp} [Acta Arith. 26, 241--251 (1975; Zbl 0263.10020)] established a law of the iterated logarithm for increasing sequences \((n_k)\) of exponential growth (``lacunary series''): \[ C_1\leq \limsup_{N \rightarrow \infty}N D_N(n_k x)(N \log \log N)^{-1/2}\leq C_2 \] for almost all real numbers \(x\), where \(D_{N}\) denotes the discrepancy of the sequence \((n_kx)\) modulo~1. Recently, \textit{K. Fukuyama} [Acta Math. Hung. 118, No. 1--2, 155--170 (2008; Zbl 1241.11090)] computed the value of the lim sup in the case \(n_k= \theta^k\), \(\theta > 1.\) Under additional arithmetic conditions on \((n_k)\) this value is equal to \(\frac{1}{2},\) the value also obtained in the case of i.i.d. sequences. In the present paper the author constructs a lacunary sequence for which the lim sup for \(D_N^{*}\) instead of \(D_N\) is not a constant and it is not equal to the lim sup for \(D_N\).
scientific article; zbMATH DE number 5702947