On effective irrationality exponents of cubic irrationals

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