Diophantine approximation and the subspace theorem
Shivani Goel, Rashi Lunia, Anwesh Ray
TL;DR
The paper presents a self-contained treatment of Roth's theorem and Schmidt’s subspace theorem, tracing from Liouville’s classical bounds to the adelic, higher-dimensional framework of the subspace theorem. It develops a toolkit of heights, Mahler measures, Siegel's lemma, and a generalized Wronskian, then constructs auxiliary polynomials and derives index bounds to obtain finiteness and structural results for approximation problems. A key thread is the adelic reformulation of Minkowski’s theorem and the use of approximation domains, height controls, and nonvanishing lemmas (including Evertse’s lemma) to generalize Roth-type estimates to several variables and derive the subspace theorem’s finiteness conclusion. The work emphasizes a streamlined, proof-oriented exposition that serves as a compact reference for the foundational results and their refinements, with implications for Diophantine equations, transcendence theory, and the distribution of rational points on varieties.
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.