Compositio Mathematica

Research Article

Uniformly Diophantine numbers in a fixed real quadratic field

Curtis T. McMullena1

a1 Mathematics Department, Harvard University, 1 Oxford St, Cambridge, MA 02138-2901, USA


The field $\mathbb {Q}(\sqrt {5})$ contains the infinite sequence of uniformly bounded continued fractions $[\overline {1,4,2,3}], [\overline {1,1,4,2,1,3}], [\overline {1,1,1,4,2,1,1,3}], \ldots ,$ and similar patterns can be found in $\mathbb {Q}(\sqrt {d})$ 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.

(Received May 14 2008)

(Revised December 16 2008)

(Accepted December 16 2008)

(Online publication April 09 2009)

2000 Mathematics Subject Classification

  • 11 (primary);
  • 37 (secondary)


  • continued fractions;
  • ideals;
  • closed geodesics;
  • packing constants


Research supported in part by the NSF.