Oppenheim continued fraction expansion and beta-expansion (Q6607329)

For \(\beta>1\) and \(x\in[0,1)\), define a map \(T_\beta:[0,1)\to[0,1)\) as \(T_\beta(x)=\beta x-\left\lfloor\beta x\right\rfloor\), where \(\left\lfloor\cdot\right\rfloor\) denotes the integer part. Then, the \(\beta\)-expansion of \(x\) is given by \(x=\sum_{n=1}^\infty\varepsilon_n(x)/\beta^n\), where \(\varepsilon_n(x)=\left\lfloor\beta T_\beta^{n-1}(x)\right\rfloor\) (\(n\ge 1\)) is the \(n\)-th digit of \(x\). When \(x=1\) and its \(\beta\)-expansion is finite, by replacing the last non-zero \(\varepsilon_m(x)\) by \(\varepsilon_m(x)-1\), we consider the infinite periodic expansion \(1=\sum_{n=1}^\infty\varepsilon_n(x)^\ast/\beta^n\) instead. Let \(\{h_n\}_{n\ge1}\) be a sequence of nonnegative integer valued functions defined on \(\mathbb N\). Let \(k_n(x)\) represent the exact number of digits in the Oppenheim continued fraction expansions of \(x\) given by the first \(n\) digits in the \(\beta\)-expansion of \(x\). Define \(l_n=\sup\{k\ge 0 : \varepsilon_{n+j}^\ast(1)=0,\, 1\le j\le k\}\). \N\NIn this paper, the authors compare the Oppenheim continued fraction expansion corresponding to the general function \(h_n\) and the \(\beta\)-expansion such that \(\beta\) satisfies the condition \(\limsup_{n\to\infty}l_n<\infty\) or \(\limsup_{n\to\infty}l_n/n=0\). Then the relations between \(k_n(x)\) and \(n\) are obtained by giving the limit, the limit inferior and the limit superior of \(k_n(x)/n\), \(k_n(x)/\sqrt{n}\) and \(k_n(x)/\log n\) depending on three conditions of \(h_n(j)\), respectively.