Metrical properties for the weighted sums of degrees of multiple partial quotients in continued fractions of Laurent series

Abstract

Let Fq((z−1)) be the field of formal Laurent series and I be the valuation ideal of Fq((z−1)). In this paper, we consider the sums of the degrees of several consecutive partial quotients raised to different powers in continued fractions over the field of Laurent series. More precisely, for m≥1 and t=(t1,t2,...,tm) with t1,t2,…,tm>0, the size of the following setFmt(ϕ)={x∈I:∑i=1mtideg⁡An+i(x)≥ϕ(n)for infinitely many n∈N} is obtained, where An(x) is the n-th partial quotient of x and ϕ(n) is a positive function defined on N.