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 \({\mathbb{Q}}\), 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 prove that the diophantine exponent of rational nilpotent Lie groups exists and is a rational number, which we determine explicitly in terms of representation theoretic data.
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Aka, M., Breuillard, E., Rosenzweig, L. et al. Diophantine approximation on matrices and Lie groups. Geom. Funct. Anal. 28, 1–57 (2018). https://doi.org/10.1007/s00039-018-0436-0
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DOI: https://doi.org/10.1007/s00039-018-0436-0