On effective irrationality exponents of cubic irrationals

Abstract

AbstractWe 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 | > 19.71 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 $$q\in \mathbb {Z}$$ q ∈ Z . These results are based on the recently discovered continued fractions of cubic irrationals and improve the currently best-known bounds of Wakabayashi.