Multidimensional generalisation of the Schmidt conjecture is true

You need to know: Euclidean space ${\mathbb R}^n$ , notation ${\mathbb N}$ for the set of positive integers. In addition, you need to know the concept of Hausdorff dimension of set $A\subset {\mathbb R}^n$ to understand the “moreover” part of the Theorem.

Background: For $x\in{\mathbb R}$ , let $||x||$ denote the distance from x to the nearest integer. Let ${\cal R}_n \subset {\mathbb R}^n$ be the set of $r=(r_1, \dots, r_n)$ such that $r_i\geq 0, \, i=1,\dots,n$ and $\sum\limits_{i=1}^n r_i=1$ . Given $\textbf{r}\in {\cal R}_n$ , we say that point $\textbf{y}=(y_1,\dots, y_n) \in {\mathbb R}^n$ is $\textbf{r}$ -badly approximable if there exists $c=c(\textbf{y})>0$ such that $\max\limits_{1 \leq i \leq n}||qy_i||^{1/r_i} \geq c/q$ for all $q \in {\mathbb N}$ , with convention that $||qy_i||^{1/0}=0$ . Let Bad(r) be be set of all $\textbf{r}$ -badly approximable points in ${\mathbb R}^n$ .

The Theorem: On 2nd April 2013, Victor Beresnevich submitted to arxiv a paper in which he proved that for any finite subset $W\subset {\cal R}_n$ , the set $\bigcap\limits_{\textbf{r}\in W} \textbf{Bad(r)}$ is non-empty. Moreover, this set has Hausdorff dimension n.

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 Bad(r) are natural generalisations containing n-tuples $(y_1, \dots, y_n)$ of simultaneously badly approximable numbers. In 1983, Schmidt conjectured that $\textbf{Bad}\left(\frac{1}{3}, \frac{2}{3}\right) \cap \textbf{Bad}\left(\frac{2}{3}, \frac{1}{3}\right) \neq \emptyset$ . In 2011, Badziahin, Pollington, and Velani proved the $n=2$ case of the Theorem, which implies the Schmidt’s conjecture as a special case. The Theorem is a generalisation of this result to all dimensions.