Rational approximations to $\sqrt[3]{2}$ and other algebraic numbers revisited

TL;DR

This paper improves irrationality measures for cube roots of integers using hypergeometric function approximations, providing near-optimal bounds and detailed arithmetic analysis of hypergeometric coefficients.

Contribution

It introduces enhanced effective irrationality measures for 33n, leveraging precise hypergeometric approximations and detailed denominator analysis.

Findings

Near-optimal irrationality bounds for 33n

Refined bounds for 33(k,l;x) and 33(k,l;x) functions

Detailed arithmetic properties of hypergeometric coefficients

Abstract

In this paper, we establish improved effective irrationality measures for certain numbers of the form $\sqrt[3]{n}$, using approximations obtained from hypergeometric functions. These results are very close to the best possible using this method. We are able to obtain these results by determining very precise arithmetic information about the denominators of the coefficients of these hypergeometric functions. Improved bounds for $\theta(k,l;x)$ and $\psi(k,l;x)$ for $k=1,3,4,6$ are also presented.