On a problem in simultaneous Diophantine approximation: Schmidt’s conjecture

Abstract

For any $i,j \ge 0$ with $i+j =1$, let $\mathbf{Bad}(i,j)$ denote the set of points $(x,y) \in \mathbb{R}^2$ for which $ \max \{ \|qx\|^{1/i}, \, \|qy\|^{1/j} \} > c/q $ for all $ q \in \mathbb{N}$. Here $c = c(x,y)$ is a positive constant. Our main result implies that any finite intersection of such sets has full dimension. This settles a conjecture of Wolfgang M. Schmidt in the theory of simultaneous Diophantine approximation.