Invariance Principles in Metric Diophantine Approximation

Michael Fuchs*

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review

2 Scopus citations

Abstract

In [7], LeVeque proved a central limit theorem for the number of solutions p, q of x-p/q ≤ f(logq)/q2 subject to the conditions 0 < q ≤ n, (p,q) < d, where x ∈ [0,1] and f satisfies certain assumptions. The case d = 1 was considerably improved a few years later by Philipp [8]. We give a common extension of both results by proving almost sure and distribution type invariance principles. Our results entail several corollaries, e.g. a functional central limit theorem and a Strassen's type version of the iterated logarithm law.

Original languageEnglish
Pages (from-to)177-203
Number of pages27
JournalMonatshefte fur Mathematik
Volume139
Issue number3
DOIs
StatePublished - 1 Jul 2003

Keywords

  • Continued fractions
  • Dependent random variables
  • Invariance principles
  • Metric diophantine approximation

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