2020/05/05 by Victor Beresnevich, Beresnevich, Victor, Erez Nesharim +3
Computer Science · Mathematics · Physics and Astronomy · #11J13 #11J83 #37A17 #Advanced Mathematical Theories and Applications #Computational Geometry and Mesh Generation #Dynamical Systems (math.DS) #FOS: Mathematics #Mathematical Dynamics and Fractals #Number Theory (math.NT)
paper · pdf · doi:10.48550/arxiv.2005.02128
openalex publication_date 2020/05/05 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
In this paper we prove that badly approximable points on any analytic non-degenerate curve in ℝn is an absolute winning set. This confirms a key conjecture in the area stated by Badziahin and Velani (2014) which represents a far-reaching generalisation of Davenport's problem from the 1960s. Amongst various consequences of our main result is a solution to Bugeaud's problem on real numbers badly approximable by algebraic numbers of arbitrary degree. The proof relies on new ideas from fractal geometry and homogeneous dynamics.