A class of algorithms for rational approximation of functions formally defined by power series

Corresponding sequence algorithms are defined and shown to exist for a wide range of corresponding continued fractions. Particular examples of these algorithms are given, including an algorithm for forming Pade approximants, and an error analysis is given in one case

Spectra of Discrete Schr\"odinger Operators with Primitive Invertible Substitution Potentials

We study the spectral properties of discrete Schr\"odinger operators with potentials given by primitive invertible substitution sequences (or by Sturmian sequences whose rotation angle has an eventually periodic continued fraction expansion, a strictly larger class than primitive invertible substitution sequences). It is known that operators from this family have spectra which are Cantor sets of zero Lebesgue measure. We show that the Hausdorff dimension of this set tends to $1$ as coupling constant $\lambda$ tends to $0$. Moreover, we also show that at small coupling constant, all gaps allowed by the gap labeling theorem are open and furthermore open linearly with respect to $\lambda$. Additionally, we show that, in the small coupling regime, the density of states measure for an operator in this family is exact dimensional. The dimension of the density of states measure is strictly smaller than the Hausdorff dimension of the spectrum and tends to $1$ as $\lambda$ tends to $0$

Beyond Expansion II: Low-Lying Fundamental Geodesics

A closed geodesic on the modular surface is "low-lying" if it does not travel "high" into the cusp. It is "fundamental" if it corresponds to an element in the class group of a real quadratic field. We prove the existence of infinitely many low-lying fundamental geodesics, answering a question of Einsiedler-Lindenstrauss-Michel-Venkatesh.Comment: 39 pages, 1 figure. This paper is a complete re-write of the posting arXiv:1310.7190, making the latter obsolete. The main tools are similar, but the application is in some sense orthogonal to the initial goal. We hope to return to the questions of the earlier paper at a later dat