Good points for diophantine approximation

Daniel Berend, Arturas Dubickas

Research output: Contribution to journalArticlepeer-review

2 Scopus citations

Abstract

Given a sequence (xn)n=1 of real numbers in the interval [0,1) and a sequence (δn)n=1 of positive numbers tending to zero, we consider the size of the set of numbers in [0,1] which can be 'well approximated' by terms of the first sequence, namely, those y ∈ [0,1] for which the inequality |y - xn| < δn holds for infinitely many positive integers n. We show that the set of 'well approximable' points by a sequence (xn)n=1, which is dense in [0,1], is 'quite large' no matter how fast the sequence (δn)n=1 converges to zero. On the other hand, for any sequence of positive numbers (δn)n=1 tending to zero, there is a well distributed sequence (xn)n=1 in the interval [0,1] such that the set of 'well approximable' points y is 'quite small'.

Original languageEnglish
Pages (from-to)423-429
Number of pages7
JournalProceedings of the Indian Academy of Sciences: Mathematical Sciences
Volume119
Issue number4
DOIs
StatePublished - 1 Sep 2009

Keywords

  • Diophantine approximation
  • Hausdorff dimension
  • Uniform distribution

ASJC Scopus subject areas

  • Mathematics (all)

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