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arXiv · 1311.5474

Dimension estimates for sets of uniformly badly approximable systems of linear forms

Ryan Broderick·Dmitry Kleinbock
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Abstract

The set of badly approximable $m \times n $ matrices is known to have Hausdorff dimension $mn $. Each such matrix comes with its own approximation constant $c$, and one can ask for the dimension of the set of badly approximable matrices with approximation constant greater than or equal to some fixed $c$. In the one-dimensional case, a very precise answer to this question is known. In this note, we obtain upper and lower bounds in higher dimensions.

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Ryan Broderick, Dmitry Kleinbock. 2013-11-21. Dimension estimates for sets of uniformly badly approximable systems of linear forms. https://arxiv.org/abs/1311.5474

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