Winning property of badly approximable points on curves

DOI10.1215/00127094-2022-0038arXiv2005.02128MaRDI QIDQ6340062

Erez Nesharim, Victor Beresnevich, Lei Yang

Publication date: 5 May 2020

Abstract: In this paper we prove that badly approximable points on any analytic non-degenerate curve in

m a t h b b R^{n}

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.

Mathematics Subject Classification ID

Simultaneous homogeneous approximation, linear forms (11J13)

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