Introduction

In 1909, a remarkable improvement on Liouville's theorem was obtained by the Norwegian mathematician Axel Thue. He proved that for any algebraic number α with degree n > 1 and for any κ > ½n + 1 there exists c = c(α, κ) > 0 such that |α–p/q| > c/qκ for all rationals p/q (q > 0). His work rested on the construction of an auxiliary polynomial in two variables possessing zeros to a high order, and it can be regarded as the source of many of our modern transcendence techniques. The condition on κ was relaxed by Siegel in 1921 to κ > s + n/(s + 1) for any positive integer s, thus, in particular, to κ > 2√n, and it was further relaxed by Dyson and Gelfond independently in 1947 to κ > √(2n). The latter expositions continued to involve polynomials in two variables and further progress seemed to require some extension of the arguments relating to polynomials in many variables; in fact special results in this connexion had already been obtained by Schneider in 1936. A generalization of the desired kind was discovered by Roth in 1955; he showed indeed that the above proposition holds for any κ > 2, a condition which, in view of the introductory remarks of Chapter 1, is essentially best possible.