Badly approximable numbers and Littlewood-type problems

We establish that the set of pairs $(\alpha ,\beta )$ of real numbers such that $$\underset{q\to +\mathrm{\infty }}{\mathrm{lim\hspace{0.17em}inf}}q\cdot (\log q{)}^2\cdot \mathrm{\parallel }q\alpha \mathrm{\parallel }\cdot \mathrm{\parallel }q\beta \mathrm{\parallel }>0,$$ where $\mathrm{\parallel }\cdot \mathrm{\parallel }$ 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