We introduce some terminology. Let
be the multiplicative group
non-zero complex numbers. Let be a subgroup of
is said to be a torsion group if all its elements have finite
order, that is, are roots of unity. In that case we say that has
More generally, is said to be
of finite rank if there are
with the following property:
there exist integers
and a positive integer such
If is not a torsion group then
the smallest for which such
exist is called the rank of .
For instance, the group
|root of unity|
First let be non-zero rational numbers and let
be the multiplicative
generated by the prime numbers
. In 1933,
Mahler  showed that the equation
has only finitely many solutions. In 1960, Lang  showed that for any and any subgroup of of finite rank, the number of solutions of equation (1) is finite.
For subgroups of there are reasonably efficient algorithms to determine all solutions of (1). For instance, consider the equation
Our concern is to give uniform upper bounds for the number of solutions
infinite classes of equations of the shape (1),
depending on as few parameters as possible.
In 1984, the author 
showed that in Mahler's case,
that is, with
and with the group generated by prime
equation (1) has at most
This bound is independent of the primes
and of the
Building further on work of Schlickewei, in 1996 Beukers and Schlickewei
 proved the following general result:
For any subgroup of of finite rank , and any , equation (1) has at most solutions.
We mention that in 1988, Erds, Stewart and Tijdeman 
result in the other direction:
Let be non-zero rational numbers. Then for every and every sufficiently large , there is a subgroup of of rank such that (1) has at least solutions.
We now turn to equations in
in , (2)
where is a subgroup of of finite rank, and . Assume that is not finite. A solution of equation (2) is called non-degenerate, if each subsum of the left-hand side is non-zero, i.e.,
It follows from work of van der Poorten and Schlickewei, the author, and
from the 1980's that (2) has only finitely many solutions. The
tool in the proof of this result is W.M. Schmidt's Subspace
(see next section).
In 1990, Schlickewei  was the first to give an explicit
upper bound for the number of non-degenerate solutions of (2),
but only in the special case that is contained in an algebraic
Schlickewei's bound depended, apart from the number of variables and
rank of , on several other parameters and when his work appeared,
it was an open problem to deduce a uniform upper bound depending only on
and the rank of .
After several intermediate results,
Schlickewei, Schmidt and the author , see also the survey
succeeded in proving the following theorem:
Theorem 1. Let be a subgroup of of finite rank , and let . Then equation (2) has at most non-degenerate solutions
The basic tool was a new quantitative version of Schmidt's Subspace Theorem, obtained by Schlickewei and the author  or the survey paper ). The upper bound in Theorem 1 is probably far from best possible, but one can show that the theorem does not remain valid if the upper bound is replaced by a bound independent of or .
We mention that recently, Moree, Stewart, Tijdeman
and the author  and independently Granville (unpublished)
proved the following generalization of the result of Erds, Stewart
Theorem 2. Let be non-zero rationals. Then for every and every sufficiently large there is a subgroup of of rank such that has at least non-degenerate solutions.
The proof is not based on Diophantine approximation but uses instead some analytic number theory.