A number theoretic question arising in the geometry of plane curves and in billiard dynamics
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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 \[ n\tan (\pi \rho )=\tan (n\pi \rho ). \] 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.References
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Additional Information
- Van Cyr
- Affiliation: Department of Mathematics, Northwestern University, Evanston, Illinois 60208
- MR Author ID: 883244
- Email: cyr@math.northwestern.edu
- Received by editor(s): March 29, 2011
- Published electronically: January 25, 2012
- Communicated by: Matthew A. Papanikolas
- © Copyright 2012
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication. - Journal: Proc. Amer. Math. Soc. 140 (2012), 3035-3040
- MSC (2010): Primary 11R18, 53A04, 37E99
- DOI: https://doi.org/10.1090/S0002-9939-2012-11258-4
- MathSciNet review: 2917076