Approximations of real numbers by the sequence \(\{\) n\(\alpha\) \(\}\) and their metrical theory
type: Item
instance of: scholarly article
title: Approximations of real numbers by the sequence \(\{\) n\(\alpha\) \(\}\) and their metrical theory
zbMATH Open document ID: 0657.10034
DOI: 10.1007/BF01952484
published in: Acta Mathematica Hungarica
publication date: 1988-01-01T00:00:00Z
review text: For an irrational real number \(0<\alpha <1\), let \([a_ 1,a_ 2,...]\) be its simple continued fraction expansion. Set \(p_ n/q_ n=[a_ 1,a_ 2,...,a_ n].\) Then any real number \(-\alpha <\beta <1-\alpha\) can uniquely be represented as \[ \beta =\sum^{+\infty}_{i=1}b_ i(\beta)(q_{i-1}\alpha -p_{i-1}), \] where, for \(b_ i=b_ i(\beta)\), the inequalities \(0\leq b_ 1\leq a_ 1-1\) and \(0\leq b_ i\leq a_ i\), \(i\geq 2\), hold (if \(b_ i=a_ i\) then \(b_{i-1}=0\) for \(i\geq 2).\) The authors prove a number of remarkable limit theorems. Two examples: (i) Set \(u_ n=\sum^{n}_{i=1}b_ i(\beta)q_{i-1}.\) Then, for almost all (\(\alpha\),\(\beta)\), \((\log u_ n)/n \to \pi^ 2/12 \log 2\) (which is known to be the (almost everywhere) limit of (log \(q_ n)/n)\). (ii) Let \(N\nu_ N\) be the number of integers \(n\leq N\) such that \[ | \beta -\sum^{n}_{i=1}b_ i(\beta)(q_{i-1}\alpha -p_{i- 1})| \leq z/q_ n. \] Then \(\nu_ N\to z(2-z)\), \(0\leq z\leq 1\).
Mathematics Subject Classification ID: 11J70
Mathematics Subject Classification ID: 11K16
zbMATH DE Number: 4073091
zbMATH Keywords: series representation
zbMATH Keywords: metric results
zbMATH Keywords: irrational real number
zbMATH Keywords: simple continued fraction expansion
zbMATH Keywords: limit theorems
author: Shunji Ito
author: Hitoshi Nakada
reviewed by: Janos Galambos
MaRDI profile type: Publication
cites work: Q5515914
cites work: Some skew product transformations associated with continued fractions and their invariant measures
cites work: On a generalization of a theorem of Borel
cites work: Some metrical theorems in number theory
cites work: Some remarks on Kuzmin's theorem for F-expansions
cites work: Irregularities of uniform distribution
description: scientific article; zbMATH DE number 4073091