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- W2324887593 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 $nge 1.$ 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 $$begin{equation*} lim _{nto infty }frac {1}{n}sum _{j=1}^n deg (A_{p_j}(alpha )) = frac {q}{q-1} end{equation*}$$ 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." @default.
- W2324887593 created "2016-06-24" @default.
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- W2324887593 date "2014-05-27" @default.
- W2324887593 modified "2023-09-23" @default.
- W2324887593 title "ON THE METRIC THEORY OF CONTINUED FRACTIONS IN POSITIVE CHARACTERISTIC" @default.
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- W2324887593 doi "https://doi.org/10.1112/s0025579314000114" @default.
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