Improved constants for effective irrationality measures from hypergeometric functions

Abstract

We simplify and improve the constant $c$ that appears in effective irrationality measures,

obtained from the hypergeometric method for $a∕b$ near $1$. The dependence of $c$ on $\left|a\right|$ in our result is best possible (as is the dependence on $n$ in many cases). For some applications, the dependence of this constant on $\left|a\right|$ becomes important. We also establish some new inequalities for hypergeometric functions that are useful in other diophantine settings.