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 x³ - tx² - 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 a4/3 | t | > 19.71 a 4 / 3 . Moreover, under the condition |t| > 86.58 a4/3 | t | > 86.58 a 4 / 3 , we provide an explicit lower bound for the expression || qx || for all large q∈ Z q ∈ Z . These results are based on the recently discovered continued fractions of cubic irrationals and improve the currently best-known bounds of Wakabayashi.