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 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 a^4/3 | t | > 19. 71 a 4 / 3. Moreover, under the condition |t| > 86. 58 a^4/3 | 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.