Continued fractions of cubic Laurent series

TL;DR

This paper constructs explicit continued fraction expansions for certain cubic Laurent series, derives bounds on rational approximations of their roots, and shows that all cubic irrationals have computable continued fractions with some having unexpectedly good rational approximations.

Contribution

It provides the first known continued fraction expansions for families of cubic Laurent series since Gauss, and establishes new bounds on rational approximations of their roots.

Findings

Constructed continued fractions for specific cubic Laurent series.

Derived lower bounds on the approximation of roots by rationals.

Identified cubic irrationals with infinitely many better-than-expected rational approximations.

Abstract

We construct continued fraction expansions for several families of the Laurent series in $\mathbb{Q}[[t^{-1}]]$. To the best of the author's knowledge, this is the first result of this kind since Gauss derived the continued fraction expansion for $(1+t)^r$, $r\in\mathbb{Q}$ in 1813. As an application, we apply an analogue of the hypergeometric method to one of those families and derive non-trivial lower bounds on the distance $|x - \frac{p}{q}|$ between one of the real roots of $3x^3 - 3tx^2-3ax+at$, $a,t\in\mathbb{Z}$ and any rational number, under relatively mild conditions on the parameters $a$ and $t$. We also show that every cubic irrational $x\in\mathbb{R}$ admits a (generalised) continued fraction expansion in a closed form that can be explicitly computed. Finally, we provide an infinite series of cubic irrationals $x$ that have arbitrarily (but finitely) many better-than-expected rational approximations. That is, they are such that for any $\tau< 3+\frac{15\ln 2}{24}\approx 3.4332...$ the inequality $||qx|| < (H(x)^{\tau} qe^{c\sqrt{\ln q}})^{-1}$ has many solutions in integer $q$.