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

For any positive integer $q\geq 2$, let $\mathbb{F}_q$ be a finite field with $q$ 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 $z$, $I$ 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 $x$. 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 $a_n(x)$. 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 $m$, 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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