Badly approximable vectors in affine subspaces: Jarn\'{\i}k-type result

TL;DR

This paper proves that in any irrational affine subspace of Euclidean space, the set of badly approximable vectors is large in the sense of winning sets, extending classical Diophantine approximation results to affine subspaces.

Contribution

It establishes that the set of badly approximable vectors in irrational affine subspaces is an $ ext{alpha}$-winning set, generalizing Jarník-type results.

Findings

The set of badly approximable vectors is $ ext{alpha}$-winning for all $ ext{alpha} o (0,1/2]$.

This extends classical Diophantine approximation results to affine subspaces.

The result implies the set has full Hausdorff dimension within the subspace.

Abstract

Consider irrational affine subspace $ A\subset \mathbb{R}^d$ of dimension $a$. We prove that the set $$ \{\xi =(\xi_1,...,\xi_d) \in {A}:\,\,\, \ q^{1/a}\cdot \max_{1\le i \le d} ||q\xi_i|| \to \infty,\,\,\,\, q\to \infty\} $$ is an $\alpha$-winning set for every $\alpha \in (0,1/2]$