scholarly journals The Lagrange spectrum of a Veech surface has a Hall ray

2016 ◽  
Vol 10 (4) ◽  
pp. 1287-1337 ◽  
Author(s):  
Mauro Artigiani ◽  
Luca Marchese ◽  
Corinna Ulcigrai
Keyword(s):  
Lagrange Spectrum ◽  
Hall Ray

scholarly journals Persistent Hall rays for Lagrange spectra at cusps of Riemann surfaces

2019 ◽  
Vol 40 (8) ◽  
pp. 2017-2072
Author(s):  
MAURO ARTIGIANI ◽  
LUCA MARCHESE ◽  
CORINNA ULCIGRAI
Keyword(s):  
Diophantine Approximation ◽  
Riemann Surfaces ◽  
Main Tool ◽  
Height Function ◽  
Classical Case ◽  
Fuchsian Groups ◽  
Proper Function ◽  
Lipschitz Norm ◽  
Lagrange Spectrum ◽  
Hall Ray

We study Lagrange spectra at cusps of finite area Riemann surfaces. These spectra are penetration spectra that describe the asymptotic depths of penetration of geodesics in the cusps. Their study is in particular motivated by Diophantine approximation on Fuchsian groups. In the classical case of the modular surface and classical Diophantine approximation, Hall proved in 1947 that the classical Lagrange spectrum contains a half-line, known as a Hall ray. We generalize this result to the context of Riemann surfaces with cusps and Diophantine approximation on Fuchsian groups. One can measure excursion into a cusp both with respect to a natural height function or, more generally, with respect to any proper function. We prove the existence of a Hall ray for the Lagrange spectrum of any non-cocompact, finite covolume Fuchsian group with respect to any given cusp, both when the penetration is measured by a height function induced by the imaginary part as well as by any proper function close to it with respect to the Lipschitz norm. This shows that Hall rays are stable under (Lipschitz) perturbations. As a main tool, we use the boundary expansion developed by Bowen and Series to code geodesics and produce a geometric continued fraction-like expansion and some of the ideas in Hall’s original argument. A key element in the proof of the results for proper functions is a generalization of Hall’s theorem on the sum of Cantor sets, where we consider functions which are small perturbations in the Lipschitz norm of the sum.

scholarly journals Markoff–Lagrange spectrum and extremal numbers

2011 ◽  
Vol 206 (2) ◽  
pp. 325-362 ◽  
Author(s):  
Damien Roy
Keyword(s):  
Lagrange Spectrum

scholarly journals On the gaps of the Lagrange spectrum

1985 ◽  
Vol 45 (1) ◽  
pp. 59-64 ◽  
Author(s):  
Bernhard Dietz
Keyword(s):  
Lagrange Spectrum

scholarly journals On the quadratic Lagrange spectrum

2013 ◽  
Vol 276 (3-4) ◽  
pp. 985-999 ◽  
Author(s):  
Yann Bugeaud
Keyword(s):  
Lagrange Spectrum

scholarly journals Quadratic Lagrange spectrum: I

2017 ◽  
Vol 289 (1-2) ◽  
pp. 515-533 ◽  
Author(s):  
Xianzu Lin
Keyword(s):  
Lagrange Spectrum

LOW HEIGHT GEODESICS ON $\Gamma^3 \backslash \mathcal{H}$: HEIGHT FORMULAS AND EXAMPLES

2007 ◽  
Vol 03 (03) ◽  
pp. 475-501 ◽  
Author(s):  
THOMAS A. SCHMIDT ◽  
MARK SHEINGORN
Keyword(s):  
Quadratic Forms ◽  
Closed Geodesics ◽  
Modular Surface ◽  
Explicit Formulas ◽  
Indefinite Quadratic ◽  
Positive Integers ◽  
Simple Closed Geodesics ◽  
Hall Ray

The Markoff spectrum of binary indefinite quadratic forms can be studied in terms of heights of geodesics on low-index covers of the modular surface. The lowest geodesics on [Formula: see text] are the simple closed geodesics; these are indexed up to isometry by Markoff triples of positive integers (x, y, z) with x2 + y2 + z2 = 3xyz, and have heights [Formula: see text]. Geodesics considered by Crisp and Moran have heights [Formula: see text]; they conjectured that these heights, which lie in the "mysterious region" between 3 and the Hall ray, are isolated in the Markoff Spectrum. In our previous work, we classified the low height-achieving non-simple geodesics of [Formula: see text] into seven types according to the topology of highest arcs. Here, we obtain explicit formulas for the heights of geodesics of the first three types; the conjecture holds for approximation by closed geodesics of any of these types. Explicit examples show that each of the remaining types is realized.

scholarly journals The Lagrange spectrum of a set

1979 ◽  
Vol 34 (4) ◽  
pp. 287-293 ◽  
Author(s):  
T. Cusick ◽  
Michel Mendès France
Keyword(s):  
Lagrange Spectrum
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