A characterization of bad approximability

We show that badly approximable vectors are exactly those that cannot, for any inhomogeneous parameter, be inhomogeneously approximated at every monotone divergent rate. This implies in particular that Kurzweil's Theorem cannot be restricted to any points in the inhomogeneous part. Our results...

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Bibliographic Details
Published in:arXiv.org 2018-01
Main Author: Ramírez, Felipe A
Format: Article
Language:English
Subjects:
Irrationality
Online Access:Get full text
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Summary:We show that badly approximable vectors are exactly those that cannot, for any inhomogeneous parameter, be inhomogeneously approximated at every monotone divergent rate. This implies in particular that Kurzweil's Theorem cannot be restricted to any points in the inhomogeneous part. Our results generalize to weighted approximations, and to higher irrationality exponents.
ISSN:2331-8422
DOI:10.48550/arxiv.1707.00771