Originally, Diophantine approximation deals with approximation of real numbers by rational numbers and with questions whether certain given numbers are irrational, algebraic, or transcendental. The basic techniques from Diophantine approximation have been vastly generalized, and there have been some striking applications to Diophantine equations and inequalities and to arithmetic algebraic geometry (11G, 14G). For instance, by means of Diophantine aproximation techniques, in 1990, G. Faltings proved the following general conjecture of S. Lang: let A be an abelian variety defined over an algebraic number field K and let X be an algebraic subvariety of A, also defined over K, which does not contain a translate of an abelian subvariety of A. Then X has only finitely many K-rational points. This is a higher dimensional generalization of the Mordell conjecture, proved by Faltings in 1983: if C is a projective curve of genus at least 2 defined over a number field K, then C has only finitely many K-rational points.

Much of my research has been related to the Subspace Theorem, proved by W.M. Schmidt in 1972:
let L₁(x) =a₁₁x₁+...+ a_1nx_n , . . . , L_n(x) =a_n1x₁+...+ a_nnx_n be linearly independent linear forms with algebraic coefficients. Let d>0. Then the set of points x=(x₁,...,x_n) in Zⁿ satisfying the inequality

(*)          |L₁(x)... L_n(x)| <  (max (|x₁|,..., |x_n|)^-d

is contained in the union of finitely many proper linear subspaces of Qⁿ.

Schmidt's proof uses Diophantine approximation techniques as well as techniques from the geometry of numbers. In 1994, Faltings and Wüstholz gave a totally different proof which avoids the geometry of numbers but which uses instead Faltings' Product Theorem. A notorious drawback of both the proofs of Schmidt and of Faltings and Wüstholz is, that they are ineffective, i.e., they do not provide an effective algorithm to find the subspaces. The Subspace Theorem is a higher dimensional generalization of Roth's theorem from 1955: for each algebraic number a and for each k> 2, there are only finitely many pairs of integers (p,q) with q>0 such that |a-(p/q)|<q^-k .

The Subspace Theorem has been extended and refined in several directions (p-adic versions, inequalities with solutions from a number field (Schmidt, Schlickewei), quantitative versions giving explicit upper bounds for the number of subspaces (Schmidt, Schlickewei, E.)). In another direction, Vojta showed that there is a finite collection of proper linear subspaces of Qⁿ, which is effectively computable and is independent of d, such that all but finitely many solutions of (*) lie in the union of these subspaces. Further, Ru and Vojta proved a "moving targets version" of the Subspace Theorem, yielding a finiteness result in a situation where also the linear forms L₁,..., L_n (the 'targets') are allowed to vary, though in a small range.

The Subspace Theorem and its generalizations and extensions have numerous applications, for instance to Diophantine inequalities (simultaneous approximation of algebraic numbers by rationals (Schmidt), approximation of algebraic numbers by algebraic numbers of bounded degree (Wirsing, Schmidt), "symmetric" approximation results implying that there are only finitely many pairs of algebraic numbers (a,b) in a given number field that are close together (E.), inequalities for resultants of two polynomials (E.)), Diophantine equations (finiteness results for S-unit equations (Schlickewei, Dubois and Rhin, van der Poorten and Schlickewei, E.), explicit upper bounds for the number of solutions of linear equations with unknowns from a multiplicative group of finite rank (Schlickewei, Schmidt, E.), norm form equations, decomposable form equations (Schmidt, Györy, Gaál, E., Ru, Berczés, Everest, Thunder), linear recurrence sequences (van der Poorten, Schlickewei, Schmidt, Corvaja, Zannier), exponential polynomial equations (Laurent, Schlickewei, Schmidt), transcendence results, e.g., for Mahler-type gap series (Nishioka, Corvaja, Zannier, Bugeaud), integral points on varieties (Corvaja, Zannier, Levin, Autissier, Vojta), and complexity of b-ary expansions of algebraic numbers and of continued fraction expansions of algebraic numbers (Adamczewski, Bugeaud).
A further development is to generalize the theory related to the Subspace Theorem to inequalities to be solved in algebraic points of a projective variety. Faltings and Wüstholz pointed out that their method to attack the Subspace Theorem can be used also to handle inequalities of the shape (*) in which the L_i are homogeneous polynomials of arbitrary degree instead of just linear forms, and moreover, the solutions x are chosen from an arbitrary projective variety. Ferretti and E. showed that this generalization of Faltings and Wüstholz can be deduced from Schmidt's Subspace Theorem and moreover, using the quantitative version of Schmidt's Subspace Theorem of Schlickewei and E., they obtained a quantitative version of the result of Faltings and Wüstholz. Similar such work has been done by Corvaja and Zannier.

In the survey Diophantine equations and Diophantine approximation which is (supposedly) meant for non-specialists I have sketched some of the consequences concerning linear equations with unknowns from a multiplicative group of finite rank and linear recurrence sequences.