Sets of inhomogeneous linear forms can be not isotropically winning

Abstract

We give an example of irrational vector $\theta \in {ℝ}^2$ such that the set

$${Bad}_{\theta }:=\left\{\left({\eta }_1,{\eta }_2\right):\underset{x\in \mathbb{N}}{inf}{x}^{1∕2}\underset{i=1,2}{max}\parallel x{\theta }_i-{\eta }_i\parallel >0\right\}$$

is not absolutely winning with respect to McMullen’s game.