Sets of inhomogeneous linear forms can be not isotropically winning

Abstract

We give an example of irrational vector $\theta \in {\mathbb{R}}^2$ such that the set $Ba{d}_{\theta }:=\{({\eta }_1,{\eta }_2):{\inf }_{x\in \mathbb{N}}{x}^{\frac{1}{2}}{\max }_{i=1,2}\Vert x{\theta }_i-{\eta }_i\Vert >0\}$ is not absolutely winning with respect to McMullen's game.

13 pages, 2 figures

References in corpus (2)

• Two-dimensional badly approximable vectors and Schmidt's game
• A note on badly approximable linear forms on manifolds