A note on badly approximable linear forms on manifolds

TL;DR

This paper investigates the size and structure of badly approximable points on manifolds within Euclidean spaces, providing new results on their dimensionality and winning properties in the context of twisted Diophantine approximation.

Contribution

It introduces two approaches to analyze badly approximable points on manifolds, establishing full dimension and isotropic winning properties under certain restrictions.

Findings

Countable intersections of badly approximable sets have full dimension.

Sets of badly approximable points are isotropically winning.

Results apply to non-degenerate C^1 submanifolds of R^n.

Abstract

This paper is motivated by Davenport's problem and the subsequent work regarding badly approximable points in submanifolds of a Euclidian space. We study the problem in the area of twisted Diophantine approximation and present two different approaches. The first approach shows that, under a certain restriction, any countable intersection of the sets of weighted badly approximable points on any non-degenerate C^1 submanifold of R^n has full dimension. In the second approach we introduce the property of isotropically winning and show that the sets of weighted badly approximable points are isotropically winning under the same restriction as above.