Badly approximable points on manifolds

TL;DR

This paper advances the understanding of badly approximable points on manifolds in higher dimensions, proving they have full dimension in intersections and establishing new techniques in Diophantine approximation.

Contribution

It develops novel methods based on lattice point counting and a key quantitative result to solve longstanding problems in Diophantine approximation on manifolds.

Findings

Any finite intersection of weighted badly approximable points on nondegenerate manifolds has full dimension.

Existence of transcendental numbers badly approximable by algebraic numbers of any degree.

New techniques applicable to higher-dimensional Diophantine approximation problems.

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.