ACHIEVEMENTS AND PROBLEMS IN DIOPHANTINE APPROXIMATION THEORY

Abstract

ContentsIntroduction I. Metrical theory of approximation on manifolds ??1. The basic problem ??2. Brief survey of results ??3. The principal conjecture II. Metrical theory of transcendental numbers ??1. Mahler's classification of numbers ??2. Metrical characterization of numbers with a given type of approximation ??3. Further problems III. Approximation of algebraic numbers by rationals ??1. Simultaneous approximations ??2. The inclusion of p-adic metrics ??3. Effective improvements of Liouville's inequality IV. Estimates of linear forms in logarithms of algebraic numbers ??1. The basic method ??2. Survey of results ??3. Estimates in the p-adic metric V. Diophantine equations ??1. Ternary exponential equations ??2. The Thue and Thue-Mahler equations ??3. Equations of hyperelliptic type ??4. Algebraic-exponential equations VI. The arithmetic structure of polynomials and the class number ??1. The greatest prime divisor of a polynomial in one variable ??2. The greatest prime divisor of a polynomial in two variables ??3. Square-free divisors of polynomials and the class number ??4. The general problem of the size of the class number Conclusion References