$\mathbf{Bad}(s,t)$ is hyperplane absolute winning

TL;DR

This paper proves that the set of badly approximable pairs with weights summing to one is hyperplane absolute winning, implying it has full Hausdorff dimension within certain fractals, strengthening previous results.

Contribution

It extends An's 2013 result by showing $ extbf{Bad}(s,t)$ is hyperplane absolute winning, enabling new dimension and intersection properties.

Findings

$ extbf{Bad}(s,t)$ is hyperplane absolute winning.

Full Hausdorff dimension within certain fractals.

Strengthens previous winning property results.

Abstract

J. An (2013) proved that for any $s,t \geq 0$ such that $s + t = 1$, $\mathbf{Bad}(s,t)$ is $(34\sqrt 2)^{-1}$-winning for Schmidt's game. We show that using the main lemma from An's paper one can derive a stronger result, namely that $\mathbf{Bad}(s,t)$ is hyperplane absolute winning in the sense of Broderick, Fishman, Kleinbock, Reich, and Weiss (2012). As a consequence one can deduce the full dimension of $\mathbf{Bad}(s,t)$ intersected with certain fractals.