Metrical properties for continued fractions of formal Laurent series

TL;DR

This paper investigates the size of sets of formal Laurent series with specific growth properties of their continued fraction partial quotients, analyzing their measure and dimension in analogy to real number theory.

Contribution

It extends the metrical theory of continued fractions to formal Laurent series, characterizing the measure and Hausdorff dimension of sets with prescribed partial quotient growth.

Findings

Determines the Haar measure of sets with growth conditions.

Calculates the Hausdorff dimension of these sets.

Provides a framework for continued fractions in formal Laurent series.

Abstract

Motivated by recent developments in the metrical theory of continued fractions for real numbers concerning the growth of consecutive partial quotients, we consider its analogue over the field of formal Laurent series. Let $A_n(x)$ be the $n$th partial quotient of the continued fraction expansion of $x$ in the field of formal Laurent series. We consider the sets of $x$ such that $\deg A_{n+1}(x)+\cdots+\deg A_{n+k}(x)~\ge~\Phi(n)$ holds for infinitely many $n$ and for all $n$ respectively, where $k\ge1$ is an integer and $\Phi(n)$ is a positive function defined on $\mathbb{N}$. We determine the size of these sets in terms of Haar measure and Hausdorff dimension.