Exact Diophantine approximation of real numbers by \(\beta\)-expansions
In this paper, the problem about the exact approximation order in the system of \(\beta\)-expansions is introduced.\N\NLet \(\beta>1\) be a real number and \(\psi:\mathbb N\to\mathbb R_{+}\) be a positive and non-increasing function defined on \(\mathbb N\) with \(\psi(n)\to 0\) as \(n\to+\infty\). Let \(E_\beta(\psi):=W(\psi)\backslash\bigcup_{0<c<1}W(c\psi)\) be the set of \(x\) which is approximable to order \(\psi\) but to no better order, where \N\[\NW(\psi):=\{x\in[0,1):x-\omega_n(x)<\psi(n)/\beta^n\,\hbox{for infinitely many}\, n\in\mathbb N\}. \N\]\NThe main result of this paper is to obtain the Hausdorff dimension of \(E_\beta(\psi)\): \N\[\N\dim_H E_\beta(\psi)=\frac{1}{1+\alpha}, \N\]\Nwhere \N\[\N\alpha:=\liminf_{n\to+\infty}\frac{-\log_\beta\psi(n)}{n}. \N\]\NThis result strengthens that in [\textit{L. Fang} et al., Math. Z. 296, No. 1--2, 13--40 (2020; Zbl 1452.11097)].
scientific article; zbMATH DE number 7896606
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