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

Abstract

We prove that for any s, t ⩾ 0 with s + t = 1 and any θ ∈ ℝ with infq∈ℕq1/s||qθ|| > 0, the set of y ∈ ℝ for which (θ, y) is (s, t)-badly approximable is ½-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.