The Schmidt conjecture in simultaneous Diophantine approximation is true

You need to know: Notations: ${\mathbb N}$ for the set of positive integers, $||x||$ for the distance from real number x to the nearest integer. In addition, you need to know the concept of Hausdorff dimension of set $A\subset {\mathbb R}^2$ to understand the context.

Background: Let S be the set of pairs of real numbers $(i,j)$ such that $0\leq i,j \leq 1$ and $i+j=1$ . For $(i; j)\in S$ , let $\text{Bad}(i; j)$ denote the set of points $(x; y)\in {\mathbb R}^2$ for which there exists a positive constant $c=c(x,y)$ such that $\max\{||qx||^{1/i}, ||qy||^{1/j}\} > c/q$ for all $q \in {\mathbb N}$ .

The Theorem: On 15th January 2010, Dzmitry Badziahin, Andrew Pollington, and Sanju Velani submitted to arxiv and the Annals of Mathematics a paper in which they proved that for any finite number $(i_1; j_1); \dots ; (i_d; j_d)$ of pairs from S, the set $\bigcap\limits_{t=1}^d \text{Bad}(i_t; j_t)$ is non-empty.

Short context: A real number x is said to be badly approximable if there exists a constant $c(x)>0$ such that $||qx|| > c(x)/q$ for all $q \in {\mathbb N}$ . Sets $\text{Bad}(i; j)$ are natural generalisations containing pairs $(x; y)$ of simultaneously badly approximable numbers. The Theorem confirms a 1983 conjecture of Schmidt. In fact, Schmidt conjectured it for specific values $d=2$ , $i_1=1/3$ , and $j_1=2/3$ , and even this remained open. Any counterexample to this conjecture would imply the famous Littlewood conjecture (see here), but the Theorem states that there is no counterexample. In fact, Badziahin, Pollington, and Velani proved a much stronger result that the set $\bigcap\limits_{t=1}^d \text{Bad}(i_t; j_t)$ is not only non-empty, but has Hausdorff dimension 2 – the same as the whole plane. In a later work, this result has been extended to higher dimensions.