Winning property of counterexamples to Uniform Littlewood's Conjecture

Abstract

In this paper, we prove that the set of counterexamples to the uniform Littlewood's conjecture proposed in Bandi-Fregoli-Kleinbock, that is, the set of pairs of real numbers $(x,y)$ satisfying $$ \limsup_{Q\to +\infty}\ Q\cdot \min_{1\leq q\leq Q}\langle qx\rangle\langle qy\rangle>0, $$ is hyperplane absolute winning. We show that a stronger statement holds: the set of real pairs $(x,y)$ satisfying $$ \liminf_{m\to +\infty}\ Q_m\cdot \min_{1\leq q\leq Q_m}\langle qx\rangle\langle qy\rangle>0, $$ is hyperplane absolute winning if $Q_{m+1} \gg Q_m^{\tau}$ for some $\tau > 1$. In particular, the above sets have full Hausdorff dimension in $\mathbb{R}^2$. In addition, we prove that these sets are absolute winning on every regular $C^2$ planar curve whose set of points of nonzero curvature is itself absolute winning on the curve. We also establish analogous results for certain lines.