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Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 

 

Cyclic approximation of irrational rotations


Author: A. Iwanik
Journal: Proc. Amer. Math. Soc. 121 (1994), 691-695
MSC: Primary 28D05; Secondary 58F11
DOI: https://doi.org/10.1090/S0002-9939-1994-1221724-X
MathSciNet review: 1221724
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Abstract: We prove that an irrational number $ \alpha $ admits a rational approximation $ \vert\alpha - p/q\vert = o(f(q))$ iff the irrational rotation $ Tx = \{ x + \alpha \} $ admits cyclic approximation with speed $ o(f(n))$. As an application to earlier results we obtain that a generic Anzai skew product over every irrational rotation is rank-1 and for a.e. $ \alpha $ most skew products admit cyclic approximation with speed $ o(1/{n^2}\log n)$.


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DOI: https://doi.org/10.1090/S0002-9939-1994-1221724-X
Article copyright: © Copyright 1994 American Mathematical Society