Functiones et Approximatio Commentarii Mathematici

A Dirichlet approximation theorem for group actions

Clayton Petsche and Jeffrey D. Vaaler

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If $G$ is a compact group acting continuously on a compact metric space $(X, m)$, we prove two results that generalize Dirichlet's classical theorem on Diophantine approximation. If $G$ is a noncommutative compact group of isometries, we obtain a noncommutative form of Dirichlet's theorem. We apply our general result to the special case of the unitary group $U(N)$ acting on the complex unit sphere, and obtain a noncommutative result in this setting.

Article information

Funct. Approx. Comment. Math., Volume 60, Number 2 (2019), 263-275.

First available in Project Euclid: 26 June 2018

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 11J25: Diophantine inequalities [See also 11D75]
Secondary: 37B05: Transformations and group actions with special properties (minimality, distality, proximality, etc.) 22F10: Measurable group actions [See also 22D40, 28Dxx, 37Axx]

unitary group continuous group actions


Petsche, Clayton; Vaaler, Jeffrey D. A Dirichlet approximation theorem for group actions. Funct. Approx. Comment. Math. 60 (2019), no. 2, 263--275. doi:10.7169/facm/1755.

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