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- Title
A DIMENSIONAL RESULT ON THE PRODUCT OF CONSECUTIVE PARTIAL QUOTIENTS IN CONTINUED FRACTIONS.
- Authors
HUANG, LINGLING; MA, CHAO
- Abstract
This paper is concerned with the growth rate of the product of consecutive partial quotients relative to the denominator of the convergent for the continued fraction expansion of an irrational number. More precisely, given a natural number $m,$ we determine the Hausdorff dimension of the following set: $$ \begin{align*} E_m(\tau)=\bigg\{x\in [0,1): \limsup\limits_{n\rightarrow\infty}\frac{\log (a_n(x)a_{n+1}(x)\cdots a_{n+m}(x))}{\log q_n(x)}=\tau\bigg\}, \end{align*} $$ where $\tau $ is a nonnegative number. This extends the dimensional result of Dirichlet nonimprovable sets (when $m=1$) shown by Hussain, Kleinbock, Wadleigh and Wang.
- Subjects
CONTINUED fractions; FRACTAL dimensions; IRRATIONAL numbers; NATURAL numbers
- Publication
Journal of the Australian Mathematical Society, 2022, Vol 113, Issue 3, p357
- ISSN
1446-7887
- Publication type
Article
- DOI
10.1017/S1446788721000173