Engel Series Expansions of Laurent Series and Hausdorff Dimensions

Abstract

For any positive integer q > 2, let F 9 be a finite field with q elements, F 9 ((z~')) be the field of all formal Laurent series x = YlT=v c ni~n in an indeterminate z, I denote the valuation ideal z^'F^tU" 1 ]] in the ring of formal power series F 9 [[z~']] and P denote probability measure with respect to the Haar measure on F 9 ((z~')) normalized by P(/) = 1. For any x e I, let the series J27=\ l/( a i0O a 2(*) • • •«*(*)) be the Engel expansion of Laurent series of*. Grabner and Knopfmacher have shown that the P-measure of the set A(a) = {x e I : limn^oodega n (x)/n = a} is 1 when a = q/{q -1), where degfl n (;c) is the degree of polynomial a n {x). In this paper, we prove that for any a > 1, A(a) has Hausdorff dimension 1. Among other things we also show that for any positive integer m, the following set B(m) = [x € / : dega n+ i(;c) -dega n (x) = m for any n > 1} has Hausdorff dimension 1.