Improved constants for effective irrationality measures from hypergeometric functions

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 | − ( κ +1) , obtained from the hypergeometric method for a/b near 1. The dependence of c on both | 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.