Diophantine approximation and the subspace theorem

TL;DR

This paper provides an accessible, self-contained exposition of Roth's theorem and the subspace theorem, highlighting their proofs and significance in Diophantine approximation and transcendence theory.

Contribution

It offers a clear, approachable presentation of classical results in Diophantine approximation, emphasizing proofs and foundational understanding.

Findings

Roth's theorem on Diophantine approximation

Schlickewei's refinement of the subspace theorem

Implications for Diophantine equations and transcendence

Abstract

Diophantine approximation explores how well irrational numbers can be approximated by rationals, with foundational results by Dirichlet, Hurwitz, and Liouville culminating in Roth's theorem. Schmidt's subspace theorem extends Roth's results to higher dimensions, with profound implications to Diophantine equations and transcendence theory. This article provides a self-contained and accessible exposition of Roth's theorem and Schlickewei's refinement of the subspace theorem, with an emphasis on proofs. The arguments presented are classical and approachable for readers with a background in algebraic number theory, serving as a streamlined, yet condensed reference for these fundamental results.