On effective irrationality exponents of cubic irrationals (Q6596371)

For an irrational number \(x\) we define its irrationality exponent as the supremum of real numbers \(\lambda\) such that the inequality \[\biggl|x-\frac{p}{q}\biggr|<q^{-\lambda}\] has infinitely many rational solutions \(p/q\).\N\NOn the other hand, the effective irrationality exponent of \(x\) is a positive real number \(\lambda_{\mathrm{eff}}(x)\) such that for all \(\lambda>\lambda_{\mathrm{eff}}(x)\) there exists an effectively computable number \(Q>0\) such that all rational solutions of the above inequality in reduced form satisfy \(q\le Q\).\N\NIn this paper, the author provides an upper bound for the effective irrationality exponent of cubic algebraic numbers. The results are based on the recently discovered continued fractions of cubic irrationals and improve the currently best-known bounds by Wakabayashi.