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

We prove that for any $s,t\ge 0$ with $s+t=1$ and any $\theta \in \mathbb{R}$ with ${\inf }_{q\in \mathbb{N}}{q}^{\frac{1}{s}}\mathrm{\parallel }q\theta \mathrm{\parallel }>0$, the set of $y\in \mathbb{R}$ for which $(\theta ,y)$ 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.