On the relative growth rate of degrees of partial quotients in continued fractions over the field of formal Laurent series

Abstract

In this paper, we study the relative growth rate of degrees of partial quotients with respect to convergents in continued fractions over the field of Laurent series. For any r > 0, the Hausdorff dimensions of the sets {x : deg An+1(x) ⩾ r · deg Qn(x) for infinitely many n} and {x : deg An+1(x) ⩾ r · deg Qn(x) for all n ⩾ 1} are determined, where An(x), n ⩾ 1, denote the partial quotients and Qn(x), n ⩾ 1, denote the denominators of the convergents.