Winning property of counterexamples to Uniform Littlewood's Conjecture

Abstract: In this paper, we prove that the set of counterexamples to uniform Littlewood's conjecture proposed in \cite{BFK25}, that is, the set of real pairs $(x,y)$ satisfying $$\limsup_{Q\to+\infty}\ Q\min_{1\leq q\leq Q}\langle qx\rangle\langle qy\rangle&gt;0$$ is hyperplane absolute winning. In particular, it has full Hausdorff dimension in $\mathbb{R}<sup>2$.