On the metric theory of continued fractions in positive characteristic (Q2874611)

Let \(\mathbb{F}_q\) be the finite field of \(q\) elements. Denote by \(\mathbb{F}_q[Z]\) and \(\mathbb{F}_q(Z)\) the ring of polynomials (with coefficients in \(\mathbb{F}_q\)) and its fraction field. The field \(\mathbb{F}_q((Z^{-1}))\) of formal Laurent series, which is non-Archimedean and of positive characteristic, is the completion of \(\mathbb{F}_q(Z)\) with respect to the valuation \(|\cdot|\) defined by \(|P/Q|:=q^{\deg (P)-\deg (Q)}\), \(P/Q \in \mathbb{F}_q(Z)\). Similar to the regular continued fractions in the field of real numbers, each \(\alpha \in \mathbb{F}_q((Z^{-1}))\) can be uniquely written as \(\alpha = A_0 + 1/(A_1+1/(A_2+ 1/(A_3+\cdots)))\), where \(A_n=A_n(\alpha)\in \mathbb{F}_q[Z]\) are polynomials with \(|A_n|>1\) for all \(n\geq 1\). This continued fraction expansion can be obtained by using a transformation \(T\) on the unit ball \(\{a_{-1}Z^{-1}+a_{-2}Z^{-2} +\cdots : a_i\in \mathbb{F}_q\}\) of \(\mathbb{F}_q((Z^{-1}))\), defined by \(T(0)=0\) and \(T(\alpha)=\{1/\alpha\}\), where \(\{a_nZ^n+\cdots +a_0+a_{-1}Z^{-1}+a_{-2}Z^{-2} +\cdots \}:=a_{-1}Z^{-1}+a_{-2}Z^{-2} +\cdots\) denotes the fractional part. In fact, we have \(A_n(\alpha)=A_0(T^n(\alpha))\).NEWLINENEWLINEThe authors first prove that the transformation \(T\) is exact with respect to the Haar measure on the unit ball of \(\mathbb{F}_q((Z^{-1}))\). Then applying some known ergodic theorems, such as the ergodic theorem for moving averages of \textit{A. Bellow} et al. [Ergodic Theory Dyn. Syst. 10, No. 1, 43--62 (1990; Zbl 0674.60035)], the authors obtain many metric properties of the continued fractions on \(\mathbb{F}_q((Z^{-1}))\). Among others, they show that for almost every \(\alpha\) with respect to the Haar measure, \(\lim_{n\to\infty} {1\over n}\sum_{j=1}^n \deg (A_{p_j}(\alpha))=q/(q-1)\), where \((p_j)_{j\geq 1}\) is the sequence of prime numbers.