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ON THE METRIC THEORY OF CONTINUED FRACTIONS IN POSITIVE CHARACTERISTIC

Part of: Probabilistic theory: distribution modulo $1$; metric theory of algorithms Measure-theoretic ergodic theory

Published online by Cambridge University Press:  27 May 2014

Poj Lertchoosakul  and
Radhakrishnan Nair
Poj Lertchoosakul
Affiliation:
Institute of Mathematics, Polish Academy of Sciences, ul. Śniadeckich 8, 00-656 Warsaw,Poland email p.lertchoosakul@impan.pl
Radhakrishnan Nair
Affiliation:
Mathematical Sciences, The University of Liverpool, Peach Street, Liverpool L69 7ZL,U.K. email nair@liv.ac.uk

Abstract

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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. Berthé and H. Nakada. Our proofs rely on pointwise subsequence and moving average ergodic theorems.

MSC classification

Secondary: 11K50: Metric theory of continued fractions 28D99: None of the above, but in this section
Type
Research Article
Information
Mathematika , Volume 60 , Issue 2 , July 2014 , pp. 307 - 320
Copyright
Copyright © University College London 2014 

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