A law of the iterated logarithm for discrepancies: non-constant limsup
type: Item
instance of: scholarly article
title: A law of the iterated logarithm for discrepancies: non-constant limsup
zbMATH Open document ID: 1215.11075
author: Katusi Fukuyama
published in: Monatshefte für Mathematik
publication date: 2010-05-17T00:00:00Z
review text: 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)].
reviewed by: Christoph Aistleitner
Mathematics Subject Classification ID: 11K38
Mathematics Subject Classification ID: 42A55
Mathematics Subject Classification ID: 60F15
zbMATH DE Number: 5708704
zbMATH Keywords: law of the iterated logarithm
zbMATH Keywords: random sequence
zbMATH Keywords: discrepancy
MaRDI profile type: Publication
full work available at URL: s00605-008-0062-2
OpenAlex ID: W1979335136
cites work: Q3505071
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DOI: 10.1007/S00605-008-0062-2
description: scientific article; zbMATH DE number 5708704