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The sets of Dirichlet non-improvable numbers versus well-approximable numbers

© 2019 Cambridge University Press

Let be a non-decreasing function, the the partial quotient of and the denominator of the the convergent. The set of -Dirichlet non-improvable numbers, \unicode[STIX]{x1D6F9}(q_{n}(x))\text{ for infinitely many }n\in \mathbb{N}\},\end{eqnarray}]]> is related with the classical set of -approximable numbers in the sense that. Both of these sets enjoy the same -dimensional Hausdorff measure criterion for. We prove that the set is uncountable by proving that its Hausdorff dimension is the same as that for the sets and. This gives an affirmative answer to a question raised by Hussain etA al [Hausdorff measure of sets of Dirichlet non-improvable numbers. MathematikaA 64(2) (2018), 502-518].

Funding

This work was first started when M. Hussain was an Endeavour research fellow at Brandeis University. M. Hussain would like to thank Professor Dmitry Kleinbock and Professor Baowei Wang for useful discussions about this problem. We would like to thank the anonymous referee for careful reading of the paper and his/her comments which improved the presentation of this paper. The research of A. Bakhtawar is supported by a La Trobe University postgraduate research award and of M. Hussain by a La Trobe University startup grant.

History

Publication Date

01/01/2019

Journal

Ergodic Theory and Dynamical Systems

Volume

40

Issue

12

Pagination

19p.

Publisher

Cambridge University Press (CUP)

ISSN

0143-3857

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