J. Aust. Math. Soc.
75 (2003), 17

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
, let
be a finite field with elements,
be the field of all formal Laurent series
in an indeterminate
, denote the valuation
ideal
in the ring of formal power series
and P
denote probability measure with respect to the
Haar measure on normalized by
. For any
, let the series


be the Engel expansion of Laurent series of
. Grabner and Knopfmacher have shown that the
Pmeasure of the set


is 1 when
, where
is the degree of polynomial
. In this paper, we prove that for any
,
has Hausdorff dimension 1. Among other things we
also show that for any positive integer
, the following set


has Hausdorff dimension 1.

Download the article in PDF format (size 66 Kb)


