Uniformly Diophantine Numbers in a Fixed Real Quadratic Field

Abstract

The field (\mathbb{Q}(\sqrt5)) 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}]), ..., 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.