Diophantine approximation and continued fraction expansion for quartic power series over $\mathbb{F}_{3}$

TL;DR

This paper explores the continued fraction expansion and irrationality measures of quartic power series over _3, extending known counterexamples to Roth's theorem in positive characteristic function fields.

Contribution

It explicitly describes a large family of quartic power series over _3 with known continued fractions and irrationality measures, extending Mahler's counterexamples.

Findings

Explicit continued fraction expansions for a family of quartic power series.

Identification of new counterexamples to Roth's theorem in positive characteristic.

Extension of the set of known algebraic power series with unbounded partial quotients.

Abstract

While Roth's theorem states that the irrationality measure of all the irrational algebraic numbers is 2, and the same holds true over function fields in characteristic zero, some counter-examples were found over function fields in positive characteristic. This was put forward first by Mahler in 1949, in his fundamental paper on Diophantine approximation \cite{M}. It seems that, except for particular elements, as power series with bounded partial quotients, Roth's theorem holds. Until now, only one element, with unbounded partial quotients, discovered by Mills and Robbins \cite{MR} in 1986, has been recognized having this property. It concerns a quartic power series over $\mathbb{F}_{3}$ having a continued fraction expansion with remarkable pattern. This continued fraction expansion was explicitly described by Buck and Robbins \cite{BR}, and later by Lasjaunias \cite{LA2} who used another method somewhat easier. Furthermore, Lasjaunias \cite{LA2} improve the value of its irrationality measure in relation with Roth's theorem. We will see that this power series is included in a large quartic power series family, for which the continued fraction expansion and the irrationality measure can be explicitly given. Moreover, we will study the rational approximation of other examples of quartic power series over $\mathbb{F}_{3}$ and we will extend the set of counter-examples initiated by Mahler.