Hausdorff dimension of singular vectors

Abstract

We prove that the set of singular vectors in ${\mathbb{R}}^d$, $d\ge 2$, has Hausdorff dimension $\frac{d^2}{d+1}$ and that the Hausdorff dimension of the set of $\epsilon$-Dirichlet improvable vectors in ${\mathbb{R}}^d$ is roughly $\frac{d^2}{d+1}$ plus a power of $\epsilon$ between $\frac{d}{2}$ and $d$. As a corollary, the set of divergent trajectories of the flow by $diag(e^t,\dots ,e^t,e^{-dt})$ acting on ${SL}_{d+1}\left(\mathbb{R}\right)/{SL}_{d+1}\left(\mathbb{Z}\right)$ has Hausdorff codimension $\frac{d}{d+1}$. These results extend the work of the first author.