A law of the iterated logarithm for discrepancies: non-constant limsup
The author solves a long-standing open problem in metric discrepancy theory and proves the existence of a sequence of integers for which the law of the iterated logarithm for the discrepancy is satisfied with a non-constant \(\limsup\). More precisely, he shows that there exists a sequence \((n_k)_{k \geq 1}\) satisfying \(1 \leq n_{k+1} - n_k \leq 5\) such that \[ \limsup_{N \to \infty} \frac{N D_N(n_k x)}{\sqrt{2 N \log \log N}} = \limsup_{N \to \infty} \frac{N D_N^*(n_k x)}{\sqrt{2 N \log \log N}} = \sigma(x) \quad \text{a.e.}, \] where \[ \sigma^2(x) = (4 x \mathbf{1}_{[0,1/4)} (x) + \mathbf{1}_{[1/4,3/4)}(x)+4(1-x)\mathbf{1}_{[3/4,1)}(x) )/9 + 1/24, \] and the same result holds for the sequence which consists of the numbers \(\mathbb{N} \backslash (n_k)_{k \geq 1}\), sorted in increasing order. A similar result for exponentially growing \((n_k)_{k \geq 1}\) has been shown independently by the reviewer [\textit{C. Aistleitner}, ``Irregular discrepancy behavior of lacunary series.'' Monatsh. Math. 160, No. 1, 1--29 (2010; Zbl 1197.11089) and ``Irregular discrepancy behavior of lacunary series. II.'' Monatsh. Math. 161, No. 3, 255--270 (2010; Zbl 1200.11057)].
scientific article; zbMATH DE number 5708704