Diophantine approximation on matrices and Lie groups
Geometric and Functional Analysis
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Aka, M., Breuillard, E., Rosenzweig, L., & Saxce, N. (2018). Diophantine approximation on matrices and Lie groups. Geometric and Functional Analysis, 28 (1) https://doi.org/10.1007/s00039-018-0436-0
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 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.
metric diophantine approximation, homogeneous dynamics, extremal manifolds, group actions
The first author acknowledges the support of ISEF, Advanced Research Grant 228304 from the ERC, and SNF Grant 200021-152819. The second author acknowledges support from ERC Grant no 617129 GeTeMo. The third author was supported by the G ̈ oran Gustafssons Stiftelse for Naturvetenskaplig och Medicinsk Forskning and Vetenskapsradet (grant no. 621-2011-5498).
External DOI: https://doi.org/10.1007/s00039-018-0436-0
This record's URL: https://www.repository.cam.ac.uk/handle/1810/275529
Attribution 4.0 International
Licence URL: http://creativecommons.org/licenses/by/4.0/
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