Uniformly Diophantine numbers in a fixed real quadratic field

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.