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Summary: Uniformly Diophantine numbers in a
fixed real quadratic field
Curtis T. McMullen
17 June, 2008
Abstract
The field Q(
5) contains the infinite sequence of uniformly bounded
continued fractions [1, 4, 2, 3], [1, 1, 4, 2, 1, 3], [1, 1, 1, 4, 2, 1, 1, 3] . . ., and
similar patterns can be found in Q(
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 man-
ifolds, packing constants of ideals, class numbers and heights.
Contents
1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
2 Lattices and quadratic fields . . . . . . . . . . . . . . . . . . . 6
3 Loop generators . . . . . . . . . . . . . . . . . . . . . . . . . . 11
4 Patterns of continued fractions . . . . . . . . . . . . . . . . . 14
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