Duke Mathematical Journal

Hausdorff dimension of singular vectors

Yitwah Cheung and Nicolas Chevallier

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We prove that the set of singular vectors in Rd, d2, 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,edt) acting on SLd+1(R)/SLd+1(Z) has Hausdorff codimension dd+1. These results extend the work of the first author.

Article information

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

Received: 20 February 2014
Revised: 13 September 2015
First available in Project Euclid: 6 September 2016

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Zentralblatt MATH identifier

Primary: 11J13: Simultaneous homogeneous approximation, linear forms 11K55: Metric theory of other algorithms and expansions; measure and Hausdorff dimension [See also 11N99, 28Dxx]
Secondary: 37A17: Homogeneous flows [See also 22Fxx]

singular vectors self-similar coverings multidimensional continued fractions simultaneous Diophantine approximation best approximations divergent trajectories


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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