Introduction

This chapter deals with Schmidt's far-reaching extension of Roth's theorem to systems of inequalities in linear forms. This is not a routine generalization; entirely new difficulties appear in the course of the proof, which Schmidt resolved by introducing new ideas from Minkowski's geometry of numbers.

In the case of systems of inequalities, it is possible to have infinitely many solutions. However, even then a finiteness theorem still holds, in the sense that solutions are contained in finitely many proper linear subspaces of the ambient space. This paves the way for applying induction arguments.

As is the case for Thue's and Roth's theorems, again Schmidt's subspace theorem is ineffective in the sense that no bound can be placed a priori on the height of the finitely many linear spaces which contain the solutions. At any rate, it remains a very flexible tool with wide applicability in many questions; the reader will find some unusual applications of the subspace theorem in this chapter.

It is also possible, as for Roth's theorem, to give an effective bound for the number of linear spaces containing the solutions. This requires rather sophisticated methods beyond the scope of this book and will not be done here.

An extension of Schmidt's theorem with a formulation allowing a finite set of places, entirely analogous to Ridout's and Lang's generalizations of Roth's theorem, was later obtained by Schlickewei. This is quite important in applications.

Section 7.2 contains several equivalent formulations of the subspace theorem.