Two-dimensional badly approximable vectors and Schmidt's game

Abstract

We prove that for any pair $(s,t)$ of nonnegative numbers with $s+t=1$, the set of two-dimensional $(s,t)$-badly approximable vectors is winning for Schmidt's game. As a consequence, we give a direct proof of Schmidt's conjecture using his game.

12 pages

References in corpus (1)

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

Cited by in corpus (8)

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• A variational principle in the parametric geometry of numbers
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• Sets of inhomogeneous linear forms can be not isotropically winning
• Winning of inhomogeneous bad for curves
• Bounded Geodesics on Locally Symmetric Spaces