Diophantine approximation in the framework of Roth’s theorem

Abstract

[en] The main goal of this work is to understand a proof of a generalized version of Roth’s theorem proposed by Lang. Due to the large scope of this proof, we will begin with older, more foundational results in Diophantine approximation, as they provide context, and introduce the general structure of the main proof in this work. Then we will study the theory of absolute values over number fields, in order to use the results and tools derived from it, such as the height functions. These functions, together with the index of a polynomial will play a huge role in the proof of the more general version of Roth’s theorem. We will then present the proof of the theorem, and finish off this work with a few applications of the theorem, as well as a discussion on an inherent limitation of the proof that carries over into other renowned theorems that depend on Roth’s theorem, such as Falting’s theorem on the finiteness of rational points in curves of genus greater or equal to two.