Badly approximable numbers and Littlewood-type problems

Yann Bugeaud, Nikolay Moshchevitin

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…