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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
Posted:
January 25, 2012
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Abstract: We prove that if  is a rational number between zero and one, then there is no integer  such that 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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H. Auerbach, Sur un problem de M. Ulam concernant l'equilibre des corps flottants. Studia. Math. 7 (1938), 121-142.
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Robert
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Csikós, Classification of first-order flexible regular
bicycle polygons, Studia Sci. Math. Hungar. 46
(2009), no. 1, 37–46. MR 2656480
(2011d:52042), http://dx.doi.org/10.1556/SScMath.2008.1074
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E. Gutkin, Capillary floating and the billiard ball problem. (2010) arXiv:1012.2448.
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E. Gutkin, Capillary floating and the billiard ball problem. J. Math. Fluid Mech. (2011). DOI 10.1007/S00021-011-0071-0.
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E. Gutkin, Billiard tables of constant width and dynamical characterizations of the circle. Abstract, Penn State Workshop, 1993.
- 6.
Patrick
Morandi, Field and Galois theory, Graduate Texts in
Mathematics, vol. 167, Springer-Verlag, New York, 1996. MR 1410264
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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
Posted:
January 25, 2012
Communicated by:
Matthew A. Papanikolas
Article copyright:
© Copyright 2012 American Mathematical Society
The copyright for this article reverts to public domain after
28 years from publication.
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