The Hausdorff dimension of the sets of irrationals with prescribed relative growth rates (Q6656085)

For a given parameter \(\vartheta\in(0,1)\), any number \(x\in(0,\vartheta)\) can be represented as a finite or infinite \(\vartheta\)-expansion \N\[\N x=[b_1(x)\vartheta,b_2(x)\vartheta,b_3(x)\vartheta,\dots]=\cfrac{1}{b_1(x)\vartheta+\cfrac{1}{b_2(x)\vartheta+\cfrac{1}{b_3\vartheta+\ddots}}}. \N\]\N The \(n\)-th (\(n\ge 1\)) convergent of the \(\vartheta\)-expansion of \(x\) is given by \N\[\N \frac{P_n(x)}{Q_n(x)}=[b_1(x)\vartheta,b_2(x)\vartheta,\dots,b_n(x)\vartheta]. \N\]\NFor every irrational \(x\in\Omega:=[0,\vartheta]\setminus\mathbb Q\), define \(\mathcal L_n(x)=\max\{b_k(x), 1\le k\le n\}\), that is, the maximum value within the initial \(n\) partial quotients block of the \(\vartheta\)-expansion of \(x\). In this paper, the authors delve into the collection of irrationals where the ratio between the largest digit and the convergents can attain a specified value. Specifically, it is shown that for any \(\alpha\ge 0\), the Haudorff dimension of \N\[\NE(\alpha)=\left\{x\in\Omega|\liminf_{n\to\infty}\frac{\mathcal L_n(x)\log\log n}{\log Q_n(x)}=\alpha\right\}.\N\]\Nis one.