Badly approximable numbers and Littlewood-type problems

Abstract

We establish that the set of pairs $(\alpha ,\beta )$ of real numbers such that $$\underset{q\to +\infty }{\mathrm{liminf}}q\cdot (\log q{)}^2\cdot \Vert q\alpha \Vert \cdot \Vert q\beta \Vert >0,$$ where $\Vert \cdot \Vert$ denotes the distance to the nearest integer, has full Hausdorff dimension in ${\mathbb{R}}^2$. Our proof rests on a method introduced by Peres and Schlag, that we further apply to various Littlewood-type problems

14 pages