Effective Irrationality Measures for Certain Algebraic Numbers

Abstract

A result of Chudnovsky concerning rational approximation to certain algebraic numbers is reworked to provide a quantitative result in which all constants are explicitly given. More particularly, Padé approximants to the function ${(1 - x)^{1/3}}$ are employed to show, for certain integers a and b, that \[ \left | {{{(a/b)}^{1/3}} - p/q} \right | > c{q^{ - \kappa }}\quad {\text {when}} q > 0.\] Here, c and k are given as functions of a and b only.