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

Let $\beta>1$, $I$ be the unite interval $[0,1)$ and $\phi$ be an integerfunction defined on $\mathbb{N}\setminus\{0\}$ satisfying $1\leq\phi(n)\leq n$.Denote by $A_\phi(x,\beta)$ the Erd\"{o}s-R\'{e}nyi average of $x\in I$associated with the function $\phi$ in $\beta$-expansion and $I_\beta$ therange of $A_\phi(x,\beta)$ for $x\in I$. For the level set \begin{align*}  ER_\phi^\beta(\alpha)=\left\{x\in I\colonA_\phi(x,\beta)=\alpha\right\},\quad\text{where}\ \alpha\in I_\beta,\end{align*} in this paper we will determine its Hausdorff dimension under theassumption $\phi(n)\to\infty$ as $n\to\infty$ and $\phi$ is the integer part ofsome slowly varying sequence. Besides, a generalization to the classic work\cite{Be} of Besicovitch is also given in $\beta$-expansion.

Hausdorff dimensions of sets related to Erd\"{o}s-R\'{e}nyi averages in beta expansions