Research Page of Thomas A. Schmidt

Some Recent Papers and Preprints:

° with C. Kraaikamp, H. Nakada Metric and arithmetic properties of mediant-Rosen maps;
   preprint, 25 pp.
              We define maps which induce mediant convergents of Rosen continued fractions and
   discuss arithmetic and metric properties of mediant convergents. In particular, we show
   equality of the ergodic theoretic Lenstra constant with the arithmetic Legendre constant
   for each of these maps. This value is sufficiently small that the mediant Rosen convergents
   directly determine the Hurwitz constant of Diophantine approximation of the underlying Fuchsian group.

° with M. Sheingorn, McShane's identity, using elliptic elements;
Geom. Ded. , vol. 134, 75-90 (2008)

        We introduce a new method to establish McShane's Identity on the weighted sum
   of the lengths of simple closed geodesics on a once-punctured hyperbolic torus. 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. The highest points of simple closed geodesics are already given as the
   appropriate complement of the regions excluded by those elements of order two that
   factor hyperbolic elements whose axis projects to be simple. The widths of the intersection
   with an appropriate horocycle of the excluded regions sum to give McShane's value of 1/2.
   The remaining points on the horocycle are highest points of simple open geodesics,
   we show that this set has zero Hausdorff dimension.

° with C. Kraaikamp, I. Smeets Tong's spectrum for Rosen continued fractions;
   J. Th. Nombres de Bordeaux, vol. 19, 641-661 (2007)
              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 coefficients. We also obtain metrical results for large blocks of ``bad''
   approximations. We use the natural extensions of Burton, Kraaikamp and Schmidt.

° with M. Sheingorn, Classifying low height geodesics on H mod Gamma^{3};
Int. J. Number Th., vol. 3, 421-438 (2007)

        We classify the topological types of all geodesics that do not penetrate far into the cusp of
     an index three cover of the modular surface.   This is directly related to the classical Markoff
     spectrum.

° with M. Sheingorn, Low height geodesics on H mod Gamma^{3}: Height formulas and examples;
Int. J. Number Th., vol. 3, 475-501 (2007)

        We proceed to identify the geodesics classified in our previous paper. In particular, we show that
     all non-simple geodesics that do not form a monogon about the cusp
     are closed and give heights of the form Sqrt[ 9 + 4/ (a_n z)^2 ], where (x, y, z) is a solution of
     Markoff's equation x^2 + y^2 + z^2 = 3 x y z, and a_n is given in terms of a recurrence relation
     depending upon z.     Replacing a_n by 1 gives the formula for the heights of the proper singly
     self-intersecting geodesics studied by Crisp and Moran in the early 1990s.

° with P. Hubert, H. Masur, A. Zorich Problems on billiards, flat surfaces and translation surfaces;
   in: Problems on mapping class groups and related topics, B. Farb, ed. Proc. Symp. Pure Math., 74. AMS (2006)
              We pose a series of questions about the matters of the title.   Extremely brief motivation and background are given.

° with P. Hubert, Geometry of infinitely generated Veech groups;
   Conformal Geometry and Dynamics, 10 (2006), p. 1-20.

         We study the surfaces constructed in our previous paper, showing that the Veech groups
          in question uniformize surfaces with both infinitely many cusps and infinitely many
          infinite ends.   The direction of any infinite end is the limit of directions of (inequivalent)
          infinite ends.

° with P. Hubert, An introduction to Veech surfaces;
Ch. 6 in: Handbook of Dynamical Systems, Vol.1B. Katok and Hasselblatt, eds. Elsevier, 2006.

             This is an elementary introduction followed by a survey of recent results.
        The introduction, focused upon the Veech Dichotomy, is based closely on lectures
        given at a workshop in Luminy, France in late June 2003.   Editors agreed to
        publish a collection of such notes, but later asked for more.   We responded with
       the survey of recent results, especially those of Calta and McMullen for the genus two
       setting, and a discussion of the constructions of infinitely generated Veech groups.

with P. Hubert, Infinitely generated Veech groups,
Duke Math. J. 123 (2004), no. 1, 49-69.

   We give a construction showing that the Veech group of a translation surface can be infinitely generated.
        This answers a question of Veech, published in 1995.    McMullen has also given infinitely generated groups,
         see our bibliography.

with E. Gutkin and P. Hubert, Affine diffeomorphisms of translation surfaces: Periodic points, Fuchsian groups, and arithmeticity
Ann. Scient. Ecole Norm. Sup., 4e ser., t. 36 (2003), 847-866.

Arising out of a pair of explicit examples identifying the location of the points of finite orbit under the action
of the affine diffeomorphisms of translation surfaces, this work shows in particular that a Veech surface is arithmetic
if and only if it has infinitely many points of finite orbit for the action.

with K. Smith, Galois orbits of principal congruence Hecke curves,
J. London Math. Soc. (2) 67 (2003), 673-685.

         We study curves defined by principal congruence subgroups of Hecke groups.     We pass from uniformizing groups to
     algebraic curves by using techniques of dessins d'enfants. Results of Streit allow us to study the action of the absolute
   Galois group on the curves that arise --- this action by way of the equations giving the canonical embedding of our
     (non-hyperelliptic) curves --- the groups acts equivariantly on the curves and on the ideals that give rise to their
     uniformizing groups.

with M.Sheingorn, Parametrizing simple closed geodesy on (Gamma^3\H),
J. Aust. Math. Soc. 74 (2003), 43-60.

The surface of the title is a degree three cover of the modular surface. So-called heights of its geodesics are
directly related to the Markoff spectrum. For each solution (x, y, z) to Markoff's equation, we associate a fundamental
    domain for the uniformizing group, with fundamental roles played by a simple closed geodesic of height Sqrt[ 9 - 4/z^2 ]
    and a paired geodesic of height Sqrt[ 9 + 4/z^2 ]. (These fundamental domains are crucial for our later work giving all
   low height geodesics on this surface.)    Furthermore, we give several descriptions of the set of simple closed geodesics,
    allowing explicit access to the subset of these of given bounded length.

with E. Savas and C. K. Koc. Generating elliptic curves of known order.
In: Cryptographic Hardware and Embedded Systems - CHES 2001, C. K. Koc, D. Naccache, and C. Paar, editors,
Lecture Notes in Computer Science No. 2162, pages 145-161, Springer Verlag, Berlin, Germany, May 13-16, 2001.

       We study a variant of the classical complex multiplication method for constructing elliptic curves of known order over
     finite fields of prime characteristic.    Heuristics for timing bounds are based upon the twin primes conjecture in imaginary
      quadratic fields.

with P. Hubert, Invariants of translation surfaces,
Ann. Inst. Fourier (Grenoble) 51 (2001), no. 2, 461-495.

        Inspired by Gutkin-Judge's result that every arithmetic Veech surface is a covering of a once-marked torus, we explore
       lattices of coverings of translation surfaces.   In particular, we introduce an invariant related to the cusps of a Teichmueller curve.
        (This invariant is at the heart of our arguments in ``The geometry of infinitely generated Veech groups.")

with P. Hubert, Veech groups and polygonal coverings
J. Geom. Phys. 35 (2000), no. 1, 75-91.

We begin our collaborative work on Veech surfaces, investigating ramified coverings of translation surfaces. We give various
examples, eg of Riemann surfaces with arbitrarily high number of 1-forms corresponding to inequivalent Veech groups. The results
inform all of our ensuing joint work.

(Filling-in to continue.)