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.

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Authors

Dzmitry Badziahin

Department of Mathematics
University of York
Heslington
York YO10 5DD
United Kingdom

Andrew Pollington

Division of Mathematical Sciences
National Science Foundation
Arlington, VA 22230

Sanju Velani

Department of Mathematics
University of York
Heslington
York YO10 5DD
United Kingdom