We give a very brief, but gentle, sketch of an introduction both to the Rosen continued fractions and to a geometric setting to which they are related, given in terms of Veech groups. We have kept the informal approach of the talk at the Numerations conference, aimed at an audience assumed to have heard of neither of the topics of the title.
The Rosen continued fractions are a family of continued fraction algorithms, each gives expansions of real numbers in terms of elements of a corresponding...

We show that Y. Cheung’s general $Z$-continued fractions can be adapted to give approximation by saddle connection vectors for any compact translation surface. That is, we show the finiteness of his Minkowski constant for any compact translation surface. Furthermore, we show that for a Veech surface in standard form, each component of any saddle connection vector dominates its conjugates in an appropriate sense. The saddle connection continued fractions then allow one to recognize certain transcendental...

We definite invariants of translation surfaces which refine Veech groups. These aid in
exact determination of Veech groups. We give examples where two surfaces of isomorphic
Veech group cannot even share a common tree of balanced affine coverings. We also show
that there exist translation surfaces of isomorphic Veech groups which cannot affinely
cover any common surface. We also extend a result of Gutkin and Judge and thereby give
the first examples of noncompact Fuchsian...

In the 1990s, J.C. Tong gave a sharp upper bound on the minimum of $k$ consecutive approximation constants for the nearest integer continued fractions. We generalize this to the case of approximation by Rosen continued fraction expansions. The Rosen fractions are an infinite set of continued fraction algorithms, each giving expansions of real numbers in terms of certain algebraic integers. For each, we give a best possible upper bound for the minimum in appropriate consecutive blocks of approximation...

We give the first transcendence results for the Rosen continued fractions. Introduced over half a century ago, these fractions expand real numbers in terms of certain algebraic numbers.

Download Results (CSV)