The method of diophantine approximation has yielded many finiteness results, as the theorems of Thue and Siegel or the theory of rational points on subvarieties of abelian schemes. Its main drawback is non-effectiveness. In the present overview I first recall some progress made in the last decade, and the remaining problems. After that I explain how to extend the known methods to some new cases, proving finiteness of integral points on certain affine schemes.

Known results

Before stating them we have to introduce some terminology. Recall that for a rational point χ ∈ ℙn(ℚ) in projective n-space we define its height as follows:

This definition can be made more sophisticated using Arakelov theory, and extends to number fields. The height measures the arithmetic complexity of the point x, and for a given bound c the number of points x with H(x) ≤ c is finite.

By restriction we get a height function on the rational points of any subvariety X ⊂ ℙn. Up to bounded functions it only depends on the ample line bundle ℒ = O (1) on X. Also one can define heights for subvarieties Z ⊂ X, or for effective algebraic cycles (see Faltings 1991).