Diophantine approximation, Khintchine's theorem, torus geometry and Hausdorff dimension

2007/10/23 by M. M. Dodson, Dodson, M. M. · 1 citation
Mathematics · Physics and Astronomy · #Advanced Mathematical Theories and Applications #FOS: Mathematics #Geometric and Algebraic Topology #Mathematical Dynamics and Fractals #Number Theory (math.NT) #math.NT

paper · pdf · doi:10.48550/arxiv.0710.4264

arxiv created 2007/10/23 · openalex publication_date 2007/10/23 · arxiv updated 2009/12/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

A general form of the Borel-Cantelli Lemma and its connection with the proof of Khintchine's Theorem on Diophantine approximation and the more general Khintchine-Groshev theorem are discussed. The torus geometry in the planar case allows a relatively direct proof of the planar Groshev theorem for the set of ψ-approximable points in the plane. The construction and use of Haudsorff measure and dimension are explained and the notion of ubiquity, which is effective in estimating the lower bound of the Hausdorff dimension for quite general lim sup sets, is described. An application is made to obtain the Hausdorff dimension of the set of approximable points in the plane when ψ(q)=q^-v, v>0, corresponding to the planar Jarnik-Besicovich theorem.

Citations

• Khintchine’s problem in metric Diophantine approximation

Cited by

• Metrical Diophantine approximation for quaternions

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