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      Permanent link:
      https://www.ias.ac.in/article/fulltext/pmsc/119/04/0423-0429

    • Keywords

       

      Uniform distribution; diophantine approximation; Hausdorff dimension.

    • Abstract

       

      Given a sequence $(x_n)^\infty_{n=1}$ of real numbers in the interval [0,1) and a sequence $(\delta_n)^\infty_{n=1}$ 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 $y\in[0,1]$ for which the inequality $|y-x_n| < \delta_n$ holds for infinitely many positive integers 𝑛. We show that the set of `well approximable’ points by a sequence $(x_n)^\infty_{n=1}$, which is dense in [0,1], is `quite large’ no matter how fast the sequence $(\delta_n)^\infty_{n=1}$ converges to zero. On the other hand, for any sequence of positive numbers $(\delta_n)^\infty_{n=1}$ tending to zero, there is a well distributed sequence $(x_n)^\infty_{n=1}$ in the interval [0,1] such that the set of `well approximable’ points 𝑦 is `quite small’.

    • Author Affiliations

       

      Daniel Berend1 Artūras Dubickas2

      1. Departments of Mathematics and Computer Science, Ben-Gurion University, Beer Sheva 84105, Israel
      2. Department of Mathematics and Informatics, Vilnius University, Vilnius LT-03225, Lithuania
    • Dates

       
  • Proceedings – Mathematical Sciences | News

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