Badly approximable points on manifolds

Abstract

This paper is motivated by two problems in the theory of Diophantine approximation, namely, Davenport's problem regarding badly approximable points on submanifolds of a Euclidean space and Schmidt's problem regarding the intersections of the sets of weighted badly approximable points. The problems have been recently settled in dimension two but remain open in higher dimensions. In this paper we develop new techniques that allow us to tackle them in full generality. The techniques rest on lattice points counting and a powerful quantitative result of Bernik, Kleinbock and Margulis. The main theorem of this paper implies that any finite intersection of the sets of weighted badly approximable points on any analytic nondegenerate submanifold of $R^n$ has full dimension. One of the consequences of this result is the existence of transcendental real numbers badly approximable by algebraic numbers of any degree.

References in corpus (5)

• Badziahin-Pollington-Velani's theorem and Schmidt's game
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• Two-dimensional badly approximable vectors and Schmidt's game
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• On a Counting Theorem of Skriganov
• Diophantine approximation on subspaces of ${\mathbb{R}}^n$ and dynamics on homogeneous spaces
• A note on badly approximable linear forms on manifolds
• A characterization of bad approximability
• Bad($\mathbf{w}$) is hyperplane absolute winning
• Weighted badly approximable vectors and games
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• Badly approximable points on manifolds and unipotent orbits in homogeneous spaces
• Bounded orbits of Diagonalizable Flows on finite volume quotients of products of $S{L}_2(\mathbb{R})$
• Some remarks on Mahler's classification in higher dimension
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• Hausdorff dimension of weighted singular vectors
• Winning of inhomogeneous bad for curves
• On the approximation exponents for subspaces of ${\mathbb{R}}^n$
• Bounded orbits of diagonalizable flows on ${\mathrm{SL}}_3(\mathbb{R})/{\mathrm{SL}}_3(\mathbb{Z})$
• Bounded Geodesics on Locally Symmetric Spaces
• Remarks on Diophantine approximation in function fields