Improved Constants for Effective Irrationality Measures from Hypergeometric Functions

TL;DR

This paper refines the constant in effective irrationality measures derived from hypergeometric functions, making it more precise and applicable, especially for values of a/b near 1, and introduces new inequalities for hypergeometric functions.

Contribution

It simplifies and improves the constant in effective irrationality measures from hypergeometric methods, with optimal dependence on |a| and n, and establishes new inequalities for hypergeometric functions.

Findings

Improved constant c in irrationality measures for hypergeometric approximations.

Optimal dependence of c on |a| and n in certain cases.

New inequalities for hypergeometric functions applicable in diophantine problems.

Abstract

In this paper, we simplify and improve the constant, $c$, that appears in effective irrationality measures, $|(a/b)^{m/n}-p/q|>c|q|^{-(\kappa+1)}$, obtained from the hypergeometric method for $a/b$ near $1$. The dependence of $c$ on $|a|$ in our result is best possible (as is the dependence on $n$ in many cases). For some applications, the dependence of this constant on $|a|$ becomes important. We also establish some new inequalities for hypergeometric functions that are useful in other diophantine settings.