Engel Series Expansions of Laurent Series and Hausdorff Dimensions

Abstract

AbstractFor any positive integer q≧2, let Fq be a finite field with q elements, Fq ((z-1)) be the field of all formal Laurent series in an inderminate z, I denote the valuation ideal z-1Fq [[z-1]] in the ring of formal power series Fq ((z-1)) normalized by P(l) = 1. For any x ∈ I, let the series be the Engel expansin of Laurent series of x. Grabner and Knopfmacher have shown that the P-measure of the set A(α) = {x ∞ I: limn→∞ deg an(x)/n = ά} is l when α = q/(q -l), where deg an(x) is the degree of polynomial an(x). In this paper, we prove that for any α ≧ l, A(α) has Hausdorff dimension l. Among other thing we also show that for any integer m, the following set B(m) = {x ∈ l: deg an+1(x) - deg an(x) = m for any n ≧ l} has Hausdorff dimension 1.