Two-dimensional badly approximable vectors and Schmidt's game

TL;DR

This paper proves that certain two-dimensional badly approximable vectors are winning in Schmidt's game, providing a direct proof of Schmidt's conjecture and advancing understanding of Diophantine approximation.

Contribution

It establishes that the set of (s,t)-badly approximable vectors is winning for Schmidt's game, confirming Schmidt's conjecture through a new direct proof.

Findings

The set of (s,t)-badly approximable vectors is winning for Schmidt's game.

A direct proof of Schmidt's conjecture is provided.

The result applies to all pairs (s,t) with s+t=1.

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.