We prove that the set of singular vectors in Rd, d≥2, has Hausdorff dimension d2d+1 and that the Hausdorff dimension of the set of ε-Dirichlet improvable vectors in Rd is roughly d2d+1 plus a power of ε between d2 and d. As a corollary, the set of divergent trajectories of the flow by diag(et,…,et,e−dt) acting on SLd+1(R)/SLd+1(Z) has Hausdorff codimension dd+1. These results extend the work of the first author.
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Cheung et al. (2016) studied this question.