Winning properties of badly approximable vectors and applications
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题目: Winning properties of badly approximable vectors and applications
摘要： In this talk, we will talk about our recent work on winning properties of badly approximable vectors. We will show that the set of weighted badly approximable vectors is hyperplane absolute winning. This confirms a conjecture by Kleinbock (1998). We also prove that weighted badly approximable points on any analytic non-degenerate curve 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. This work is joint with Victor Beresnevich and Erez Nesharim.