On the Quantitative Metric Theory of Continued Fractions in Positive Characteristic

Poj Lertchoosakul,Radhakrishnan Nair

doi:10.1017/s0013091517000177

Abstract

AbstractLet 𝔽q be the finite field of q elements. An analogue of the regular continued fraction expansion for an element α in the field of formal Laurent series over 𝔽q is given uniquely by $$\alpha = A_0(\alpha ) + \displaystyle{1 \over {A_1(\alpha ) + \displaystyle{1 \over {A_2(\alpha ) + \ddots }}}},$$ where $(A_{n}(\alpha))_{n=0}^{\infty}$ is a sequence of polynomials with coefficients in 𝔽q such that deg(An(α)) ⩾ 1 for all n ⩾ 1. In this paper, we provide quantitative versions of metrical results regarding averages of partial quotients. A sample result we prove is that, given any ϵ > 0, we have $$\vert A_1(\alpha ) \ldots A_N(\alpha )\vert ^{1/N} = q^{q/(q - 1)} + o(N^{ - 1/2}(\log N)^{3/2 + {\rm \epsilon }})$$ for almost everywhere α with respect to Haar measure.

Highlights

Let Fq denote the finite field of q elements, where q is a power of a prime p
We summarize the contents of this paper
The following three theorems can be viewed as corollaries of Theorem 1. We note that they should be compared with Theorems 12–14 of [7] as they sharpen those results when∞ j=1 is considered as the sequence of natural numbers in the literature

Summary

Introduction

Let Fq denote the finite field of q elements, where q is a power of a prime p. They proved the positive characteristic analogue of Khinchin’s famous result that lim. Specializing for instance to the case F (x) = logq x, we establish the quantitative version of the positive characteristic Khinchin’s constant for almost everywhere α ∈ B(0; 1) with respect to Haar measure, [2, 6]. We note that they should be compared with Theorems 12–14 of [7] as they sharpen those results when (aj)∞ j=1 is considered as the sequence of natural numbers in the literature

Lemmas

Proofs