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

Abstract

We prove that for any $s,t\ge 0$ with $s+t=1$ and any $\theta \in \mathbb{R}$ with ${\inf }_{q\in \mathbb{N}}{q}^{\frac{1}{s}}\Vert q\theta \Vert >0$, 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.

12 pages

References in corpus (1)

• Badly approximable vectors on a vertical Cantor set

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• Badly approximable points on manifolds and unipotent orbits in homogeneous spaces
• Bounded orbits of certain diagonalizable flows on $S{L}_n(\mathbb{R})/S{L}_n(\mathbb{Z})$
• Hausdorff dimension of weighted singular vectors
• Badly approximable points on planar curves and a problem of Davenport
• Bounded Geodesics on Locally Symmetric Spaces
• Geometric considerations around the Littlewood conjecture
• Bounded orbits of diagonalizable flows on ${\mathrm{SL}}_3(\mathbb{R})/{\mathrm{SL}}_3(\mathbb{Z})$