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** Previous:** Diophantine Equations and Diophantine

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.