ON THE METRIC THEORY OF CONTINUED FRACTIONS IN POSITIVE CHARACTERISTIC

Abstract

Let $\def \xmlpi #1{}\def \mathsfbi #1{\boldsymbol {\mathsf {#1}}}\let \le =\leqslant \let \leq =\leqslant \let \ge =\geqslant \let \geq =\geqslant \def \Pr {\mathit {Pr}}\def \Fr {\mathit {Fr}}\def \Rey {\mathit {Re}}\mathbb{F}_q$ be the finite field of $q$ elements. An analogue of the regular continued fraction expansion for an element $\alpha $ in the field of formal Laurent series over $\mathbb{F}_q$ is given uniquely by $$\begin{equation*} \alpha = A_0(\alpha )+\cfrac {1}{A_1(\alpha )+\cfrac {1}{A_2(\alpha )+\ddots }}, \end{equation*}$$ where $(A_n(\alpha ))_{n=0}^\infty $ is a sequence of polynomials with coefficients in $\mathbb{F}_q$ such that $\deg (A_n(\alpha ))\ge 1$ for all $n\ge 1.$ We first prove the exactness of the continued fraction map in positive characteristic. This fact implies a number of strictly weaker properties. Particularly, we then use the weak-mixing property and ergodicity to establish various metrical results regarding the averages of partial quotients of continued fraction expansions. A sample result that we prove is that if $(p_n)_{n=1}^\infty $ denotes the sequence of prime numbers, we have $$\begin{equation*} \lim _{n\to \infty }\frac {1}{n}\sum _{j=1}^n \deg (A_{p_j}(\alpha )) = \frac {q}{q-1} \end{equation*}$$ for almost every $\alpha $ with respect to Haar measure. In the case where the sequence $(p_n)_{n=1}^\infty $ is replaced by $(n)_{n=1}^\infty ,$ this result is due to V. Houndonougbo, V. Berthe and H. Nakada. Our proofs rely on pointwise subsequence and moving average ergodic theorems.