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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

A number theoretic question arising in the geometry of plane curves and in billiard dynamics


Author: Van Cyr
Journal: Proc. Amer. Math. Soc. 140 (2012), 3035-3040
MSC (2010): Primary 11R18, 53A04, 37E99
Published electronically: January 25, 2012
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Abstract: We prove that if $ \rho \neq 1/2$ is a rational number between zero and one, then there is no integer $ n>1$ such that

$\displaystyle n\tan (\pi \rho )=\tan (n\pi \rho ). $

This proves a conjecture due to E. Gutkin which he formulated in connection with mathematical billiards. It also may be viewed as a rigidity result for the circle in the theory of bicycle curves.

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Additional Information

Van Cyr
Affiliation: Department of Mathematics, Northwestern University, Evanston, Illinois 60208
Email: cyr@math.northwestern.edu

DOI: http://dx.doi.org/10.1090/S0002-9939-2012-11258-4
PII: S 0002-9939(2012)11258-4
Keywords: Cyclotomic field, bicycle curve, mathematical billiards.
Received by editor(s): March 29, 2011
Published electronically: January 25, 2012
Communicated by: Matthew A. Papanikolas
Article copyright: © Copyright 2012 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.