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The generalised Hausdorff measure of sets of Dirichlet non-improvable numbers.
- Source :
-
Proceedings of the American Mathematical Society . May2023, Vol. 151 Issue 5, p1823-1838. 16p. - Publication Year :
- 2023
-
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. [ABSTRACT FROM AUTHOR]
- Subjects :
- *HAUSDORFF measures
*REAL numbers
*INTEGERS
Subjects
Details
- Language :
- English
- ISSN :
- 00029939
- Volume :
- 151
- Issue :
- 5
- Database :
- Academic Search Index
- Journal :
- Proceedings of the American Mathematical Society
- Publication Type :
- Academic Journal
- Accession number :
- 162264414
- Full Text :
- https://doi.org/10.1090/proc/16222