Continued fractions of cubic Laurent series

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 $q$.