Abstract
We prove that badly approximable points on any analytic nondegenerate curve in Rn are an absolute winning set. This confirms a key conjecture in the area stated by Badziahin and Velani in 2014 that represents a far-reaching generalization of Davenport’s problem from the 1960s. Among 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.
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