## Duke Mathematical Journal

### Hausdorff dimension of singular vectors

#### Abstract

We prove that the set of singular vectors in $\mathbb{R}^{d}$, $d\ge2$, has Hausdorff dimension $\frac{d^{2}}{d+1}$ and that the Hausdorff dimension of the set of $\varepsilon$-Dirichlet improvable vectors in $\mathbb{R}^{d}$ is roughly $\frac{d^{2}}{d+1}$ plus a power of $\varepsilon$ between $\frac{d}{2}$ and $d$. As a corollary, the set of divergent trajectories of the flow by $\operatorname{diag}(e^{t},\dots,e^{t},e^{-dt})$ acting on $\operatorname{SL}_{d+1}(\mathbb{R})/\operatorname{SL}_{d+1}(\mathbb{Z})$ has Hausdorff codimension $\frac{d}{d+1}$. These results extend the work of the first author.

#### Article information

Source
Duke Math. J. Volume 165, Number 12 (2016), 2273-2329.

Dates
Revised: 13 September 2015
First available in Project Euclid: 6 September 2016

https://projecteuclid.org/euclid.dmj/1473186401

Digital Object Identifier
doi:10.1215/00127094-3477021

Mathematical Reviews number (MathSciNet)
MR3544282

Zentralblatt MATH identifier
1358.11078

#### Citation

Cheung, Yitwah; Chevallier, Nicolas. Hausdorff dimension of singular vectors. Duke Math. J. 165 (2016), no. 12, 2273--2329. doi:10.1215/00127094-3477021. https://projecteuclid.org/euclid.dmj/1473186401

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