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arXiv · 1603.03800

Diophantine approximation on matrices and Lie groups

Menny Aka·Emmanuel Breuillard·Lior Rosenzweig·Nicolas de Saxcé
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Abstract

We study the general problem of extremality for metric Diophantine approximation on submanifolds of matrices. We formulate a criterion for extremality in terms of a certain family of algebraic obstructions and show that it is sharp. In general, the almost sure diophantine exponent of a submanifold is shown to depend only on its Zariski closure, and when the latter is defined over the rational numbers, we prove that the exponent is rational and give a method to effectively compute it. This method is applied to a number of cases of interest, in particular, we manage to determine the diophantine exponent of random subgroups of certain nilpotent Lie groups in terms of representation theoretic data.

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Menny Aka, Emmanuel Breuillard, Lior Rosenzweig, Nicolas de Saxcé. 2017-12-02. Diophantine approximation on matrices and Lie groups. https://arxiv.org/abs/1603.03800

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