Engel series expansions of Laurent series and Hausdorff dimensions

J. Aust. Math. Soc. 75 (2003), 1-7

Engel series expansions of Laurent series and Hausdorff dimensions

Jun Wu
Department of Mathematics
Wuhan University
Wuhan, Hubei, 430072
People's Republic of China
wujunyu@public.wh.hb.cn

Abstract

For any positive integer $q\geq 2$ , let $\mathbb{F}_q$ be a finite field with

elements, $\mathbb{F}_q((z^{-1}))$ be the field of all formal Laurent series $x=\sum^\infty_{n=v}c_n z^{-n}$ in an indeterminate

denote the valuation ideal $z^{-1}\mathbb{F}_q[[z^{-1}]]$ in the ring of formal power series $\mathbb{F}_q[[z^{-1}]]$ and P denote probability measure with respect to the Haar measure on $\mathbb{F}_q((z^{-1}))$ normalized by ${\bf P}(I)=1$ . For any $x \in I$ , let the series

$\sum_{n=1}^{\infty}1/(a_1(x)a_2(x)\cdots a_n(x))$

be the Engel expansion of Laurent series of

. Grabner and Knopfmacher have shown that the P-measure of the set

$A(\alpha)= \{x \in I: \lim_{n \to \infty} \operatorname{deg}a_n(x)/n = \alpha \}$

is 1 when $\alpha=q/(q-1)$ , where $\operatorname{deg} a_n(x)$ is the degree of polynomial

. In this paper, we prove that for any $\alpha \geq 1$ , $A(\alpha)$ has Hausdorff dimension 1. Among other things we also show that for any positive integer

, the following set

$B(m)=\{x \in I: \operatorname{deg}a_{n+1}(x)-\operatorname{deg}a_n(x) =m \text{ for any } n \geq 1\}$

has Hausdorff dimension 1.

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