  In this paper we prove that badly approximable points on any analytic
non-degenerate curve in $\mathbb{R}^n$ 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.
