Littlewood's conjecture

Let $(\alpha,\beta)$ be a pair of real numbers, and let $\epsilon>0$. The system

$$\begin{cases} |p+q\alpha||r+q\beta|<\epsilon/T\\ q\leq T \end{cases}$$

uses solutions $(p,r,q)\in\mathbb{Z}\times\mathbb{Z}\times\mathbb{N}$ for an unbounded set of $T>0$. Littlewood's conjecture. For every pair $(\alpha,\beta)$ and every $\epsilon>0$, there is an unbounded set of $T>0$ for which the system has a solution. Almost every pair satisfies this property, and the set of multiplicatively badly approximable pairs has zero Hausdorff dimension, but the conjecture remains unresolved.