Transcendence with Rosen continued fractions
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.
math.NT↗
arXiv subjects
Thomas A. Schmidt
Publications and source records attributed to Thomas A. Schmidt.
Low height geodesics and the Markoff spectrum
We classify, in terms of topology of highest arcs, low height non-simple geodesics on the modular hyperbolic punctured sphere with three elliptic fixed points of order two. Of eight possible types, exactly one consists of geodesics that form a bigon about the cusp; we express all such geodesics in terms of Markoff triples. We also show that any geodesic joining two elliptic fixed points has locally isolated height.
math.GT↗
Veech surfaces with non-periodic directions in the trace field
We show that each of Veech's original examples of translation surfaces with ``optimal dynamics'' whose trace field is of degree greater than two has non-periodic directions of vanishing SAF-invariant. Furthermore, we give explicit examples of pseudo-Anosov diffeomorphisms whose contracting direction has zero SAF-invariant.
math.DS↗
Quilting natural extensions for alpha-Rosen Fractions
We give natural extensions for the alpha-Rosen continued fractions of Dajani et al. for a set of small alpha values by appropriately adding and deleting rectangles from the region of the natural extension for the standard Rosen fractions. It follows that the underlying maps have equal entropy.
math.NT↗
Metric and arithmetic properties of mediant-Rosen maps
A continued fractions based verification of the Hurwitz values for the Hecke triangle groups is given, completing a program of Lehner's. Ergodic theory shows that Diophantine approximation by mediant convergents of the Rosen continued fractions is sufficient to determine the values that Haas and Series found by hyperbolic geometry.
math.NT↗
McShane's identity, using elliptic elements
We introduce a new method to establish McShane's Identity, based upon the fact that elliptic elements of order two in the Fuchsian group uniformizing the quotient of a fixed once-punctured hyperbolic torus act so as to exclude points as being highest points of geodesics.
math.MG↗
Tong's spectrum for Rosen continued fractions
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 coefficients (in the sense of Diophantine approximation by continued fraction convergents). We also obtain metrical results for large blocks of ``bad'' approximations.
math.NT↗
Geometry of infinitely generated Veech groups
Veech groups uniformize Teichmüller geodesic curves in Riemann moduli space. Recently, examples of infinitely generated Veech groups have been given. We show that these can even have infinitely many cusps and infinitely many infinite ends. We further show that examples exist for which each direction of an infinite end is the limit of directions of inequivalent infinite ends.
math.GT↗