Explicit lower bounds for rational approximation to algebraic numbers

Abstract

In this paper, we apply Padé approximation methods to derive completely explicit measures of irrationality for certain classes of algebraic numbers. Our approach is similar to that taken previously by G.V. Chudnovsky but has some fundamental advantages with regards to determining implicit constants. Our general results may be applied to produce specific bounds of the flavour of

$$ \left| \sqrt[3]{2} - \frac{p}{q} \right| > \frac{1}{4}~ q^{-2.45} \hskip2ex \mbox{and} \hskip2ex \left| \sqrt[7]{5} - \frac{p}{q} \right| > \frac{1}{4}~ q^{-4.43} $$

which we show to hold for any nonzero integers $p$ and $q$. Further examples are tabulated and applications to Diophantine equations are briefly discussed as are other topics of related interest.

1991 Mathematics Subject Classification: primary 11J68, 11J82; secondary 11D41.