On the relative growth rate of the product of consecutive partial quotients in continued fraction expansions 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)). For any x∈I, let x=[A1(x),A2(x),…] be its continued fraction expansion and let Pn(x)/Qn(x) be the n-th convergents of x. Given m≥1, for any α>0, we defineDm(α):={x∈I:∏i=n+1n+m‖Ai(x)‖≥‖Qn(x)‖α for ultimatelyn∈N} andFm(α):={x∈I:∏i=n+1n+m‖Ai(x)‖≥‖Qn(x)‖α for infinitely manyn∈N}, where ‖⋅‖ represents the norm of a polynomial in Fq((z−1)). In this paper, we study the size of the above sets from the perspective of Hausdorff dimension.