On effective irrationality exponents of cubic irrationals

Abstract

We provide an upper bound for the effective irrationality exponents of cubic algebraics x with the minimal polynomial \(x^3 - tx^2 - a\). In particular, we show that it becomes non-trivial, i.e. better than the classical bound of Liouville, in the case \(|t| > 19.71 a^{4/3}\). Moreover, under the condition \(|t| > 86.58 a^{4/3}\), we provide an explicit lower bound for the expression ||qx|| for all large \(q\in \mathbb {Z}\). These results are based on the recently discovered continued fractions of cubic irrationals and improve the currently best-known bounds of Wakabayashi.