Badly approximable points on manifolds

By Victor Beresnevich. 2013

65 citations recorded by the source corpus.

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 $$\mathbb {R}^n$$Rn has full dimension. One of the consequences of this result is the existence of transcendental real numbers badly approximable by algebraic numbers of any bounded degree.

Subfields: Computational Theory and Mathematics, Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/346dba9084c0179bb8676b57617b81788d7ee82d

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In Computational Theory and Mathematics

In Geometry and Topology

In Mathematical Physics

Citations in the corpus

Exponents of Diophantine approximation (2006, background): “ry results from [31], the answer to Problem 9.8 is positive for some integer n ≥ 4 if one can prove that any real transcendental number ξ satisﬁes wˆn−1(ξ) = n− 1. Despite the recent progress made in [4, 8], the following problem remains open. 29 Problem 9.9. There exist a real transcendental number”
Metric Diophantine Approximation : aspects of recent work (2016, methodology): “The framework of Cantor-rich sets was utilised in the same paper [11] to establish the following result concerning badly approximable points on manifolds.”
Assouad Dimension and Fractal Geometry (2020, background): “The setting where F is a manifold is motivated by work of Davenport [50](2) and has developed into a broad research area, see [27] for some background.”