Sets of inhomogeneous linear forms can be not isotropically winning

TL;DR

The paper constructs an example of an irrational vector in two dimensions for which a specific set related to inhomogeneous Diophantine approximation is not absolutely winning, challenging assumptions about the size of such sets.

Contribution

It provides the first explicit example showing that certain inhomogeneous approximation sets are not absolutely winning, revealing limitations of existing winning set properties.

Findings

The set $Bad_{\pmb{\theta}}$ is not absolutely winning.

This challenges previous beliefs about the size and structure of inhomogeneous approximation sets.

The example is explicit and specific to a constructed irrational vector.

Abstract

We give an example of irrational vector $\pmb{\theta} \in \mathbb{R}^2$ such that the set $Bad_{\pmb{\theta}} := \{(\eta_1,\eta_2): \inf_{x\in\mathbb{N}} x^{\frac{1}{2}} \max_{i=1,2} \|x \theta_i-\eta_i\|>0\}$ is not absolutely winning with respect to McMullen's game.