The dimension of well approximable numbers

2025/09/06 by Victor Beresnevich, Beresnevich, Victor, Sanju Velani +1
Computer Science · #11J13 #11J71 #11J83 #11K55 #11K60 #28A78 #28A80 #42B10 #43A46 #Computability, Logic, AI Algorithms #FOS: Mathematics #Number Theory (math.NT)

paper · pdf · doi:10.48550/arxiv.2509.05687

openalex publication_date 2025/09/06 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

In this survey article, we explore a central theme in Diophantine approximation inspired by a celebrated result of Besicovitch on the Hausdorff dimension of well approximable real numbers. We outline some of the key developments stemming from Besicovitch's result, with a focus on the Mass Transference Principle, Ubiquity and Diophantine approximation on manifolds and fractals. We highlight the subtle yet profound connections between number theory and fractal geometry, and discuss several open problems at their intersection.

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Related

• Lüroth Expansions in Diophantine Approximation: Metric Properties and Conjectures
• Hausdorff Dimension and Diophantine Approximation
• Hausdorff and packing dimension of Diophantine sets
• Mass transference principle from rectangles to rectangles in Diophantine approximation
• Metric Diophantine Approximation: aspects of recent work