We start with some history. Let be a real irrational algebraic
number
of degree
and let .
In 1909, Thue [28] proved that
for any
, the inequality

(6)

has only finitely many solutions in pairs of integers with
.
After improvements of Thue's result by Siegel, Gel'fond and Dyson,
in 1955 Roth [22] proved that
(6) has only finitely many solutions
in pairs of integers with already when .
This lower bound is best possible, since by a result of Dirichlet
from 1842,
for any irrational
real number there are infinitely many pairs of integers
with

In a sequence of papers from 1965-1972, W.M. Schmidt proved a far
reaching higher
dimensional generalization of Roth's theorem, now known as the Subspace
Theorem.
For a full proof of the Subspace Theorem as well as of the other results
mentioned
above we refer to Schmidt's lecture notes [25]. Below we
have stated
the version of the Subspace Theorem which is most convenient for us.
We define the norm of
by
.

**Subspace Theorem (Schmidt).**
*Let
*

to be solved simultaneously in integer vectors .

Then there are proper linear subspaces of such that the set of solutions of (7) is contained in .

Roth's Theorem follows by taking , , , , . Thus, if is a solution of (6) with then also satisfies (7).

We give another example to illustrate the Subspace Theorem.
Consider the system

(8)

The Pell equation
has infinitely many solutions in
positive
integers . It is easy to see that if is a solution
of the
Pell equation with
and if , then
is a
solution of (8). Thus, the subspace contains
infinitely many
solutions of (8).
One can prove something more precise than predicted by the Subspace
Theorem,
that is, that (8) has only finitely many solutions with
.

In 1977, Schlickewei [23] proved a so-called p-adic version
of the Subspace Theorem, involving, apart from the usual absolute value,
a finite
number of p-adic absolute values. Given a rational number
and a prime number , we define
where is
the exponent
such that
with integers not divisible by
.
For instance, and
. The -adic absolute value
defines a metric on
. By taking the metric completion
we obtain a field
. Let
denote the algebraic closure of
.
The -adic absolute value can be extended
uniquely to
. To get a uniform notation, we write
for the usual absolute value , and
for
.
We
call the infinite prime of
. We will use the index to
indicate
either or a prime number.
Then we get:
**p-adic Subspace Theorem (Schlickewei).**
*Let
consist of the infinite prime
and a
finite number of primes numbers. For , let
*

to be solved simultaneously in .

Then there are proper linear subspaces of such that the set of solutions of (9) is contained in .

There is a further generalization of this result, which we shall not state, dealing with systems of inequalities to be solved in vectors consisting of integers from a given algebraic number field. This generalization has a wide range of applications, such as finiteness results for Diophantine equations of the type considered in the previous sections, finiteness results for all sorts of Diophantine inequalities, transcendence results, finiteness results for integral points on surfaces, etc.

As an illustration, we consider the equation

(10)

to be solved in
.
It is easy to see that
(10) has only solutions with non-negative
.
Notice that
is a solution of
in
.
Hence equation (10) may be viewed as a special case of (2).

Put
,
,
,
,
,
.
Then
and
.
Hence there are
such that
and
. We consider those solutions
with fixed values of . Notice that these solutions satisfy the
inequalities

This system is a special case of (9), and since the sum of the exponents is we can apply the p-adic Subspace Theorem with .

Taking into consideration the possibilities for , we see that is contained in the union of finitely many proper linear subspaces of . Considering the solutions in a single subspace, we can eliminate one of the variables and obtain an equation of the same type as (10), but in only two variables. Applying again the p-adic Subspace Theorem but now with , we obtain that the solutions lie in finitely many one-dimensional subspaces, etc. Eventually we obtain that (10) has only finitely many solutions.

In 1989, Schmidt [26] obtained a quantitative version of his Subspace Theorem, giving an explicit upper bound for the number of subspaces . Since then, his result has been refined and improved in several directions. In particular Schlickewei obtained quantitative versions of his p-adic Subspace Theorem which enabled him to prove weaker versions of Theorem 1 with an upper bound depending on and other parameters and of Schmidt's theorem on linear recurrences with an upper bound depending on and other parameters. Finally, Schlickewei and the author [7] managed to prove a quantitative version of the p-adic Subspace Theorem with unknowns taken from the ring of integers of a number field which was strong enough to imply the upper bounds mentioned in the previous sections. We will not give the rather complicated statement of this result.

By using a suitable specialization argument
from algebraic geometry
one may reduce
Theorem 1 to the case that
and the group are
contained
in an algebraic number field,
and then subsequently one may reduce equation (2)
to a finite number of systems (9) by a similar argument as
above.
By applying the quantitative p-adic Subspace Theorem to each of these
systems
and adding together the upper bounds for the number
of subspaces for each system,
one obtains an explicit upper bound
for the number of subspaces containing the solutions of (2).
Considering the solutions
of (2) in one
of these subspaces, then by eliminating one of the variables one obtains
an equation of the shape (2) in variables to which a
similar
argument can be applied. By repeating this, Theorem 1 follows.

The proof of Schmidt's theorem on linear recurrence sequences has a
similar structure,
but there the argument is much more involved.