## Abstract

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

Original language | English |
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Pages (from-to) | 423-429 |

Number of pages | 7 |

Journal | Proceedings of the Indian Academy of Sciences: Mathematical Sciences |

Volume | 119 |

Issue number | 4 |

DOIs | |

State | Published - 1 Sep 2009 |

## Keywords

- Diophantine approximation
- Hausdorff dimension
- Uniform distribution

## ASJC Scopus subject areas

- Mathematics (all)