Badziahin-Pollington-Velani's theorem and Schmidt's game

We prove that for any $s,t\ge0$ with $s+t=1$ and any $θ\in\mathbb{R}$ with $\inf_{q\in\mathbb{N}}q^{\frac{1}{s}}\|qθ\|>0$, the set of $y\in\mathbb{R}$ for which $(θ,y)$ is $(s,t)$-badly approximable is 1/2-winning for Schmidt's game. As a consequence, we remove a technical assumption in a recent theorem of Badziahin-Pollington-Velani on simultaneous Diophantine approximation.

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