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<p>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 result to higher dimensions, with profound implications for Diophantine equations and transcendence theory. This article provides a self-contained and accessible exposition of Roth’s theorem and Schlickewei’s refinement of Schmidt’s 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.</p>

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Diophantine Approximation and the Subspace Theorem

  • 期刊论文

  • 发表日期：

    2026年8月2日

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  • Shivani Goel,
  • Rashi Lunia,
  • Anwesh Ray

摘要

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 result to higher dimensions, with profound implications for Diophantine equations and transcendence theory. This article provides a self-contained and accessible exposition of Roth’s theorem and Schlickewei’s refinement of Schmidt’s 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.