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

TL;DR

This paper proves that certain sets related to badly approximable numbers are winning in Schmidt's game, removing a technical assumption in a recent theorem on Diophantine approximation.

Contribution

It establishes the 1/2-winning property of specific sets in Schmidt's game for all relevant parameters, extending previous results.

Findings

Sets are 1/2-winning in Schmidt's game.

Removes technical assumption in recent Diophantine approximation theorem.

Enhances understanding of badly approximable sets in Diophantine theory.

Abstract

We prove that for any $s,t\ge0$ with $s+t=1$ and any $\theta\in\mathbb{R}$ with $\inf_{q\in\mathbb{N}}q^{\frac{1}{s}}\|q\theta\|>0$, 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.