English (United Kingdom)

https://curated-unify.zendy.io/wp-json/zendy-region/v1/featured_content/oa?rat=en

https://curated-unify.zendy.io/wp-json/zendy-region/v1/highlighted_journal/

Global: Zendy Plus annual

Presents the access of premium content as premium feature

Premium Content

Presents the keyphrase highlighting as premium feature

Keyphrase Highlighting

Presents the summarisation as premium feature

Summarisation

Insights

Presents the pdf analysis as premium feature

PDF Analysis

Presents the zaia usage as premium feature

ZAIA

Global: Zendy Plus monthly

Global: Zendy Tools annual

Global: Zendy Tools monthly

Global: Zendy Free Plan

For a real $x\in(0,1)\setminus\mathbb{Q}$, let $x=[a_1(x),a_2(x),\cdots]$ beits continued fraction expansion. Denote by $T_n(x):= max \{a_k(x): 1\leq k\leqn\}$ the leading partial quotient up to $n$. For any real $\alpha\in(0,\infty),\gamma\in(0,\infty)$, let $F(\gamma,\alpha):=\{x\in(0,1)\setminus\mathbb{Q}:\lim_{n\rightarrow\infty}\frac{T_n(x)}{e^{n^\gamma}}=\alpha\}$. For a set$E\subset (0,1)\setminus\mathbb{Q}$, let $dim_H E$ be its Hausdorff dimension.Recently Lingmin Liao and Michal Rams [LR, Theorem 1.3] show that $dim_HF(\gamma,\alpha)$ is $1$ if $r\in(0,1/2)$, it is $1/2$ if $r\in(1/2,\infty)$for any $\alpha\in(0,\infty)$. In this paper we show that $dim_HF(1/2,\alpha)=1/2$ for any $\alpha\in(0,\infty)$ following Liao and Rams'method, which supplements their result.

A remark on Liao and Rams' result on distribution of the leading partial quotient with growing speed $e^{n^{1/2}}$ in continued fractions