On effective irrationality exponents of cubic irrationals

Abstract

We provide an upper bound on the efficient irrationality exponents of cubic algebraics $x$ with the minimal polynomial $x^3-t{x}^2-a$. In particular, we show that it becomes non-trivial, i.e. better than the classical bound of Liouville in the case $|t|>19.71a^{4/3}$. Moreover, under the condition $|t|>86.58a^{4/3}$, we provide an explicit lower bound on 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.

17 pages