We start with some history. Let be a real irrational algebraic
and let .
In 1909, Thue  proved that
, the inequality
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  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 . 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
We give another example to illustrate the Subspace Theorem.
Consider the system
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  proved a so-called p-adic version
of the Subspace Theorem, involving, apart from the usual absolute value,
number of p-adic absolute values. Given a rational number
and a prime number , we define
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
denote the algebraic closure of
The -adic absolute value can be extended
. To get a uniform notation, we write
for the usual absolute value , and
call the infinite prime of
. We will use the index to
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
As an illustration, we consider the equation
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
In 1989, Schmidt  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  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
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
By applying the quantitative p-adic Subspace Theorem to each of these
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
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.