The generalised Hausdorff measure of sets of Dirichlet non-improvable numbers
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- by Philip Bos, Mumtaz Hussain and David Simmons
- Proc. Amer. Math. Soc. 151 (2023), 1823-1838
- DOI: https://doi.org/10.1090/proc/16222
- Published electronically: February 28, 2023
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Abstract:
Let $\psi :\mathbb {R}_+\to \mathbb {R}_+$ be a non-increasing function. A real number $x$ is said to be $\psi$-Dirichlet improvable if the system \begin{equation*} |qx-p|< \psi (t) \ \ {\text {and}} \ \ |q|<t \end{equation*} has a non-trivial integer solution for all large enough $t$. Denote the collection of such points by $D(\psi )$. In this paper, we prove a zero-infinity law valid for all dimension functions under natural non-restrictive conditions. Some of the consequences are zero-infinity laws, for all essentially sublinear dimension functions proved by Hussain-Kleinbock-Wadleigh-Wang [Mathematika 64 (2018), pp. 502–518], for some non-essentially sublinear dimension functions, and for all dimension functions but with a growth condition on the approximating function.References
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Bibliographic Information
- Philip Bos
- Affiliation: Department of Mathematical and Physical Sciences, La Trobe University, Bendigo 3552, Australia
- MR Author ID: 309035
- Email: phil@philbos.com
- Mumtaz Hussain
- Affiliation: Department of Mathematical and Physical Sciences, La Trobe University, Bendigo 3552, Australia
- MR Author ID: 858704
- ORCID: 0000-0001-5621-9341
- Email: M.Hussain@latrobe.edu.au
- David Simmons
- Affiliation: Department of Mathematics, The University of York, England, United Kingdom
- MR Author ID: 1005497
- Email: David.Simmons@york.ac.uk
- Received by editor(s): October 20, 2020
- Received by editor(s) in revised form: February 24, 2022, June 23, 2022, and July 13, 2022
- Published electronically: February 28, 2023
- Additional Notes: The second and third named authors were supported by the Australian Research Council Discovery Project (ARC DP200100994). The third named author was a Royal Society University Research Fellow.
- Communicated by: Matthew A. Papanikolas
- © Copyright 2023 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 151 (2023), 1823-1838
- MSC (2020): Primary 11J83, 11K60
- DOI: https://doi.org/10.1090/proc/16222
- MathSciNet review: 4556181