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

Cyr, Van

doi:10.1090/S0002-9939-2012-11258-4

A number theoretic question arising in the geometry of plane curves and in billiard dynamics
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Proc. Amer. Math. Soc. 140 (2012), 3035-3040 Request permission

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.

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