On effective irrationality exponents of cubic irrationals

TL;DR

This paper establishes new upper bounds on the irrationality exponents of certain cubic algebraic numbers, improving upon classical bounds and providing explicit lower bounds under specific conditions, based on recent continued fraction developments.

Contribution

It introduces improved bounds on irrationality exponents for cubic irrationals with specific minimal polynomials, leveraging recent continued fraction techniques.

Findings

Upper bounds better than Liouville's for certain parameters

Explicit lower bounds on approximation quality for large denominators

Improves upon previous bounds by Wakabayashi

Abstract

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