We introduce some terminology. Let be the multiplicative group of 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 rank 0. More generally, is said to be of finite rank if there are with the following property: for every there exist integers and a positive integer such that . 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 group 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 of 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 numbers , equation (1) has at most solutions. This bound is independent of the primes and of the coefficients . 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  proved a 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 variables, namely
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 Laurent from the 1980's that (2) has only finitely many solutions. The major tool in the proof of this result is W.M. Schmidt's Subspace Theorem (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 number field. Schlickewei's bound depended, apart from the number of variables and the 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 paper  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 and Tijdeman mentioned above:
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.