In number theory the result from the paper by J. Liouville from 1844 [“Sur des classes trés étendues dont la valeur n’est ni algébriques, ni même réductibles des irrationelles algébriques”, C. R. Acad. Sci Paris 18, 883–885 (1844)] has been ‘lifted’ to continued fractions with bounded partial quotients and later the search for results like that for \(p\)-adic continued fractions have been pursued.
The paper under review focuses its attention on continued fractions in \(\mathbb{Q}_p,\ p\) prime. Let
1. \((\alpha_i)_{i\geq 1}\in\mathbb{N}^{\ast},\ \alpha_0\in\mathbb{Z}\) and \((b_i)_{i\geq 0}\in\{1,\ldots,p-1\}\),
2. \((n_k)_{k\geq 0},\ (\lambda_k)_{k\geq 0}\) and \((r_k)_{k\geq 0}\) are sequences of positive integers,
3. \(n_{k+1}\geq n_k+\lambda_kr_k\) for all \(k\geq 0\),
Then a \(p\)-adic number \(\xi\) is said to be a so-called quasi-periodic Schneider continued fraction if it is is given by \[ \xi=\left[(\alpha_0,b_0),\ldots,\underset{\leftarrow}{(\alpha_{n_0},b_{n_0})},\underset{\lambda_0}{\ldots},\underset{\rightarrow}{(\alpha_{n_0+r_0-1},b_{n_0+r_0-1})},\ldots,\underset{\leftarrow}{(\alpha_{n_k},b_{n_k})},\underset{\lambda_k}{\ldots},\underset{\rightarrow}{(\alpha_{n_k+r_k-1},b_{n_k+r_k-1})},\ldots\right], \] where \(\alpha_{m+r_k}=\alpha_m,\ b_{m+r+k}=b_m\) for \(n_k\leq m\leq n_k+(\lambda_k-1)r-k-1\).
The main result of the paper is now given by
Theorem 3.1 Let a quasi-periodic Schneider continued fraction \(\xi\) as defined above be given and let \((\alpha_k,b_k)_{k\geq 0}\) be a sequence that is not ultimately periodic. Suppose that \((\alpha_k)_{k\geq 0}\) is bounded and set \(A=\max\,\{\alpha_k|k\in \mathbb{N}\}\).
Assume that \(\alpha_{n_k}=\alpha_{n_k+1}=\cdots =\alpha_{n_k+r_k-1}=\alpha\) for infinitely many \(k\geq 0\). If \[ \limsup_{k\rightarrow +\infty}\,\frac{\lambda_k r_k}{n_k}>1, \] then \(\xi\) is either quadratic or transcendental.
The layout of the paper is as follows
§1. Introduction (\(1\) page)
§2. Schneider continued fraction in \(\mathbb{Q}_p\) (\(1\frac{1}{2}\) pages)
§3. Main results (\(4\frac{1}{2}\) pages)
References (\(16\) items)
A very nice paper.