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

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.