Xuehai Hu and Luming Shen Huazhong Agricultural University, Science College of Hunan Agricultural University

Abstract : Let $\mathbb{F}_q$ be a finite field with $q$ elements and $\mathbb{F}_q((X^{-1}))$ be the field of all formal Laurent series with coefficients lying in $\mathbb{F}_q$. This paper concerns with the size of the set of points $x\in \mathbb{F}_q((X^{-1}))$ with their partial quotients $A_n(x)$ both lying in a given subset $\mathbb{B}$ of polynomials in $\mathbb{F}_q[X]$ ($\mathbb{F}_q[X]$ denotes the ring of polynomials with coefficients in $\mathbb{F}_q$) and $\deg A_n(x)$ tends to infinity at least with some given speed. Write $ E_{\mathbb{B}}=\{x: A_n(x)\in \mathbb{B}, \deg A_n(x)\to \infty \ {\rm{as}}\ n\to \infty\}. $ It was shown in [8] that the Hausdorff dimension of $ E_{\mathbb{B}}$ is $\inf\{s: \sum_{b \in \mathbb{B}}(q^{-2\deg b})^{s}<\infty\}. $ In this note, we will show that the above result is sharp. Moreover, we also attempt to give conditions under which the above dimensional formula still valid if we require the given speed of $\deg A_n(x)$ tends to infinity.

Keywords : continued fractions, Laurent series, partial quotient, Hausdorff dimension