On continued fraction expansions of quadratic irrationals in positive characteristic

TL;DR

The paper studies continued fraction expansions of quadratic irrationals in positive characteristic, focusing on how the coefficients behave under polynomial scaling such as $P^n f$ and along Hecke rays. Employing a dynamical framework that connects the Artin map with the diagonal flow on a moduli space of lattices via a cross-section, it establishes a robust link between $A$-orbits and $\Psi$-orbits, and leverages Hecke-tree techniques and Kem–Pau–Sha results to prove degree-escaping behavior and irregular distributions. The main contributions include (i) a $c$-degree-escaping phenomenon for coefficients in $P^n f$ and along Hecke rays, (ii) a precise measure-theoretic correspondence between invariant measures on the cross-section and $\Psi$-invariant measures on the Artin dynamics, and (iii) a distribution result showing non-absolute-continuity of limiting measures for uncountably many Hecke directions, highlighting stark contrasts with real-characteristic zero phenomena. The results illuminate a rich arithmetic-dynamical structure in positive characteristic and provide a framework for understanding large-deviation patterns in continued fractions of quadratic irrationals over function fields.

Abstract

Let $P$ be a prime polynomial in the variable $Y$ over a finite field and let $f$ be a quadratic irrational in the field of formal Laurant series in the variable $Y^{-1}$. We study the asymptotic properties of the degrees of the coefficients of the continued fraction expansion of quadratic irrationals such as $P^nf$ and prove results that are in sharp contrast to the analogue situation in zero characteristic.

Theorems & Definitions (6)

• Theorem 1
• Theorem 2
• Theorem 3
• Theorem 4
• Proposition 5
• Lemma 6