Quantitative versions of the Subspace Theorem and applications

The Subspace Theorem, whose name will be clear from its statement, was proved by Wolfgang Schmidt around forty years ago. It provides us with a multidimensional extension of Roth’s Theorem and was originally developed for the study of two classical problems, namely algebraic approximation to algebraic numbers and norm form equations (a class of Diophantine equations which includes the Thue equations). Subsequent applications (of suitable extensions of the Subspace Theorem) to unit equations and linear recurrence sequences were published some ten years later, as well as a proof of a conjecture of Lang and many further applications to families of Diophantine equations which include norm form equations. During the last decade, several quite unexpected applications of the Subspace Theorem were found, some of which have been discussed by Yuri Bilu in his talk at the Séminaire Bourbaki [14]. These include new transcendence criteria, finiteness results for the number of solutions to families of exponential Diophantine equations, and the work of Corvaja and Zannier (with subsequent developments by Autissier and Levin) on integral points on curves and surfaces. Roughly speaking, the Subspace Theorem asserts that, for every n ≥ 2, all the integral solutions (x1, . . . , xn) to a given system of linear equations with real algebraic coefficients are, under some necessary conditions, contained in a finite union S1 ∪ . . . ∪ St of proper rational subspaces of Q. Like Roth’s Theorem, it is ineffective, in the sense that its proof does not give an upper bound for the height of the subspaces S1, . . . ,St. However, Schmidt was able in 1989 to establish an explicit upper bound for t, the number of proper rational subspaces which contain all the solutions. Such a statement is called the Quantitative Subspace Theorem. One of the purposes of this expository text is to show the importance of this quantitative statement and to display some of its (sometimes rather unexpected) consequences. In Sections 2 and 3 we state the Subspace Theorem and discuss some of its most classical applications. Section 4 is concerned with several recent applications, mainly to transcendence theory. We continue in Section 5 with a statement of a version of the Quantitative Subspace Theorem and mention its applications to equations with unknowns in a finitely generated multiplicative group and to linear recurrence sequences. In Section 6, we show how a quantitative version of Roth’s Theorem can be used to improve, under an extra hypothesis, the conclusion of Roth’s Theorem itself in two different directions. We further explain that similar ideas can be worked out with the Quantitative Subspace The-