Metrical properties for continued fractions of formal Laurent series (Q2031652)

It's well known that every irrational number \(x \in (0, 1)\) can be uniquely expressed as a simple continued fraction expansion as follows \( x:=[a_1(x),a_2(x),a_3(x),\dots] \) where \(a_n (x)\) are positive integers. Let \(\mathbb{F}_q\) be a finite field with \(q\) elements and \(\mathbb{F}_q((z^{-1}))\) denotes the field of all formal Laurent series \(x=\sum_{n=\nu}^{\infty} c_n z^{-n} \) with coefficients \(c_n \in \mathbb{F}_q \). Let \(I\) be the valuation ideal of \(\mathbb{F}_q((z^{-1})) \), that is, \[ I=\{x \in \mathbb{F}_q((z^{-1})) : |x|_{\infty} <1 \}=\left\{ \sum_{n=1}^{\infty} c_n z^{-n} : c_n \in \mathbb{F}_q \right\}. \] It's well known that each \( x \in I\) has a finite or infinite continued fraction expansion induced by the Gauss transformation, \( x := [A_1(x),A_2(x),\dots], \) where the partial quotients \(A_i(x) \) are polynomials of a strictly positive degree. Let \(\Phi: \mathbb{N} \to (1,\infty)\) be a positive function. The set \[ \mathcal{F}_k(\Phi) :=\left\{ x \in I : \sum_{i=1}^k \deg A_{n+i}(x) \geq \Phi(n) \; \text{for infititely many} \; n \in \mathbb{N} \right\} \] is defined. The \(\nu\)-measure and Hausdorff dimension of the set \(\mathcal{F}_k(\Phi)\) calculated in the paper. The size of the following set \[ \mathcal{G}_k(\Phi) :=\left\{ x \in I : \sum_{i=1}^k \deg A_{n+i}(x) \geq \Phi(n) \; \text{for all} \; n \in \mathbb{N} \right\} \] is obtained too.