Introduction

Originally, Diophantine approximation is the branch of number theory dealing with problems such as whether a given real number is rational or irrational, or whether it is algebraic or transcendental. More generally, for a given irrational number one may ask how well it is approximable by a rational number, and for a given transcendental number one may ask how well it can be approximated by algebraic numbers. The basic techniques from Diophantine approximation have been vastly generalized and today, there are some very powerful results with many applications, in particular to Diophantine equations. In this note we will discuss linear equations whose unknowns are taken from a multiplicative group of finite rank. The results we will mention about these equations are consequences of a central theorem in Diophantine approximation, the so-called Subspace Theorem of W.M. Schmidt. We will also give some results on linear recurrence sequences. In the last section we will mention some recent developments in Diophantine geometry.