Badly Approximable Systems of Affine Forms and Incompressibility on Fractals

TL;DR

This paper investigates the Hausdorff dimension of certain exceptional sets related to badly approximable systems of linear forms, demonstrating their strong C^1 incompressibility using a variant of Schmidt's game.

Contribution

It introduces a new application of Schmidt's game to establish the C^1 incompressibility of badly approximable sets and their images on fractals.

Findings

The set of badly approximable systems is strongly C^1 incompressible.

Diffeomorphic images of these sets retain their Hausdorff dimension.

Refined techniques improve estimates of exceptional sets' dimensions.

Abstract

We explore and refine techniques for estimating the Hausdorff dimension of exceptional sets and their diffeomorphic images. Specifically, we use a variant of Schmidt's game to deduce the strong C^1 incompressibility of the set of badly approximable systems of linear forms as well as of the set of vectors which are badly approximable with respect to a fixed system of linear forms.