Dimension of the set of singular vectors

Singular vectors were defined by Khintchin in the twenties. Recently it has been proved by Yitwah Cheung that the Hausdorff dimension of the set of singular couples is $4/3$. In a joint work with Yitwah Cheung, we have proved that the Hausdorff dimension of the set of singular vectors in $\mathbb R^d$ is $\frac{d^2}{d+1}$.
We will explain the proof of this formula with a special emphasis on best Diophantine approximations.
This talk is supported by “Short-time visits of foreign scientists to Russia” of Dynastia Foundation