lagrange spectrum
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Author(s):  
Byungchul Cha ◽  
Heather Chapman ◽  
Brittany Gelb ◽  
Chooka Weiss
Keyword(s):  


2021 ◽  
Vol 380 ◽  
pp. 107547
Author(s):  
Shigeki Akiyama ◽  
Hajime Kaneko


2020 ◽  
Vol 95 (3) ◽  
pp. 593-633
Author(s):  
Carlos Matheus ◽  
Carlos Gustavo Moreira


2020 ◽  
Vol 358 (4) ◽  
pp. 475-479
Author(s):  
Carlos Matheus
Keyword(s):  


Author(s):  
Davi Lima ◽  
Carlos Matheus ◽  
Carlos Gustavo Moreira ◽  
Sandoel Vieira

Abstract We show that $1+3/\sqrt{2}$ is a point of the Lagrange spectrum $L$ that is accumulated by a sequence of elements of the complement $M\!\setminus\! L$ of the Lagrange spectrum in the Markov spectrum $M$. In particular, $M\!\setminus\! L$ is not a closed subset of $\mathbb{R}$, so that a question by T. Bousch has a negative answer.



2019 ◽  
Vol 40 (8) ◽  
pp. 2017-2072
Author(s):  
MAURO ARTIGIANI ◽  
LUCA MARCHESE ◽  
CORINNA ULCIGRAI

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.



2019 ◽  
Vol 8 (1) ◽  
pp. 47-56
Author(s):  
Dmitry Gayfulin
Keyword(s):  


2018 ◽  
Vol 225 (2) ◽  
pp. 553-607 ◽  
Author(s):  
Pascal Hubert ◽  
Samuel Lelièvre ◽  
Luca Marchese ◽  
Corinna Ulcigrai
Keyword(s):  


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


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