Hostname: page-component-78c5997874-lj6df Total loading time: 0 Render date: 2024-11-13T00:42:05.051Z Has data issue: false hasContentIssue false
  • English
  • Français

Uniformly Diophantine numbers in a fixed real quadratic field

Published online by Cambridge University Press:  01 July 2009

Curtis T. McMullen
Curtis T. McMullen*
Affiliation:
Mathematics Department, Harvard University, 1 Oxford St, Cambridge, MA 02138-2901, USA
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

The field contains the infinite sequence of uniformly bounded continued fractions and similar patterns can be found in for any d>0. This paper studies the broader structure underlying these patterns, and develops related results and conjectures for closed geodesics on arithmetic manifolds, packing constants of ideals, class numbers and heights.

Keywords

continued fractionsidealsclosed geodesicspacking constants
Type
Research Article
Information
Compositio Mathematica , Volume 145 , Issue 4 , July 2009 , pp. 827 - 844
Copyright
Copyright © Foundation Compositio Mathematica 2009

References

[1]Borevich, Z. I. and Shafarevich, I. R., Number theory (Academic Press, New York, 1966).Google Scholar
[2]Cassels, J. W. S. and Swinnerton-Dyer, H. P. F., On the product of three homogeneous linear forms and the indefinite ternary quadratic forms, Philos. Trans. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci. 248 (1955), 73–96.Google Scholar
[3]Crutchfield, J. P., Farmer, J. D., Packard, N. H. and Shaw, R. S., Chaos, Scientific American 255 (1986), 46–57.CrossRefGoogle Scholar
[4]Duke, W., Hyperbolic distribution problems and half-integral weight Maass forms, Invent. Math. 92 (1988), 73–90.CrossRefGoogle Scholar
[5]Dyson, F. J. and Falk, H., Period of a discrete cat mapping, Amer. Math. Monthly 99 (1992), 603–614.CrossRefGoogle Scholar
[6]Einsiedler, M., Lindenstrauss, E., Michel, P. and Venkatesh, A., The distribution of periodic torus orbits on homogeneous spaces. Preprint, 2006.Google Scholar
[7]Ghys, E., Variations autour du théorème de récurrence de Poincaré, J. de maths des élèves (l’ENS de Lyon) 1 (1994), 3–12.Google Scholar
[8]Gruber, P. M. and Lekkerkerker, C. G., Geometry of Numbers (Elsevier, Amsterdam, 1987).Google Scholar
[9]Hardy, G. H. and Wright, E. M., An introduction to the theory of numbers (Oxford University Press, Oxford, 1979).Google Scholar
[10]Hensley, D., A polynomial time algorithm for the Hausdorff dimension of continued fraction Cantor sets, J. Number Theory 58 (1996), 9–45.CrossRefGoogle Scholar
[11]Jarnik, V., Zur metrischen Theorie der diophantischen Approximationen, Prace Mat.-Fiz. 36 (1928), 91–106.Google Scholar
[12]Katok, S. and Ugarcovici, I., Symbolic dynamics for the modular surface and beyond, Bull. Amer. Math. Soc. 44 (2007), 87–132.CrossRefGoogle Scholar
[13]Lagarias, J. C., On the computational complexity of determining the solvability or unsolvability of the equation X 2−DY 2=−1, Trans. Amer. Math. Soc. 260 (1980), 485–508.Google Scholar
[14]Lang, S., Elliptic functions (Springer-Verlag, Berlin, 1987).CrossRefGoogle Scholar
[15]Linnik, Yu. V., Ergodic properties of algebraic fields (Springer-Verlag, Berlin, 1968).CrossRefGoogle Scholar
[16]Margulis, G. A., Problems and conjectures in rigidity theory, in Mathematics: frontiers and perspectives (American Mathematical Society, Providence, RI, 2000), 161–174.Google Scholar
[17]Nathanson, M. B. and Sullivan, B. D., Heights in finite projective space, and a problem on directed graphs, Integers 8 (2008), A13, 9 pp.Google Scholar
[18]Niederreiter, H., Dyadic fractions with small partial quotients, Monatsh. Math. 101 (1986), 309–315.CrossRefGoogle Scholar
[19]Pollicott, M., Distribution of closed geodesics on the modular surface and quadratic irrationals, Bull. Soc. Math. France 114 (1986), 431–446.CrossRefGoogle Scholar
[20]Sands, J. W., Generalization of a theorem of Siegel, Acta Arith. 58 (1991), 47–57.CrossRefGoogle Scholar
[21]Sarnak, P., Class numbers of indefinite binary quadratic forms, J. Number Theory 15 (1982), 229–247.CrossRefGoogle Scholar
[22]Schmidt, W. M., Badly approximable systems of linear forms, J. Number Theory 1 (1969), 139–154.CrossRefGoogle Scholar
[23]Series, C., Geometric methods of symbolic coding, in Ergodic theory, symbolic dynamics, and hyperbolic spaces (Oxford University Press, Oxford, 1991), 125–152.Google Scholar
[24]Shallit, J., Real numbers with bounded partial quotients: a survey, Enseign. Math. 38 (1992), 151–187.Google Scholar
[25]Smale, S., Diffeomorphisms with many periodic points, in Differential and combinatorial topology (Princeton University Press, Princeton, NJ, 1965), 63–80.CrossRefGoogle Scholar
[26]Wilson, S. M. J., Limit points in the Lagrange spectrum of a quadratic field, Bull. Soc. Math. France 108 (1980), 137–141.CrossRefGoogle Scholar
[27]Woods, A. C., The Markoff spectrum of an algebraic number field, J. Aust. Math. Soc. A 25 (1978), 486–488.CrossRefGoogle Scholar
[28]Zaremba, S. K., La méthode des ‘bons treillis’ pour le calcul des intégrales multiples, in Applications of number theory to numerical analysis, Proceedings Symposia, University of Montreal, Montreal, Quebec, 1971 (Academic Press, New York, 1972), 39–119.CrossRefGoogle Scholar