Intrinsic Diophantine approximation on the unit circle and its Lagrange spectrum

Cha, Byungchul; Kim, Dong Han

doi:10.5802/aif.3522

Cha, Byungchul ¹; Kim, Dong Han ²

¹ Muhlenberg College 2400 Chew st Allentown, PA 18104 (USA)
² Department of Mathematics Education Dongguk University - Seoul 30 Pildong-ro 1-gil, Jung-gu Seoul 04620 (Korea)

Annales de l'Institut Fourier, Volume 73 (2023) no. 1, pp. 101-161.

Abstract (OV)
Résumé

Let $ℒ (S^{1})$ be the Lagrange spectrum arising from intrinsic Diophantine approximation on the unit circle $S^{1}$ by its rational points. We give a complete description of the structure of $ℒ (S^{1})$ below its smallest accumulation point. To this end, we use digit expansions of points on $S^{1}$ , which were originally introduced by Romik in 2008 as an analogue of simple continued fraction of a real number. We prove that the smallest accumulation point of $ℒ (S^{1})$ is 2. Also we characterize the points on $S^{1}$ whose Lagrange numbers are less than 2 in terms of Romik’s digit expansions. Our theorem is the analogue of the celebrated theorem of Markoff on badly approximable real numbers.

Soit $ℒ (S^{1})$ le spectre de Lagrange associé à l’approximation diophantienne intrinsèque sur le cercle unité $S^{1}$ par ses points rationnels. Nous donnons une description complète de la structure de $ℒ (S^{1})$ en dessous de son plus petit point d’accumulation. Pour cela, nous utilisons les développements infinis des points de $S^{1}$ introduits en 2008 par Romik, comme analogues des développements en fraction continue des nombres réels. Nous montrons que 2 est le plus petit point d’accumulation de $ℒ (S^{1})$ . Nous caractérisons également les développements de Romik des points de $S^{1}$ dont les nombres de Lagrange sont inférieurs à 2. Notre théorème est l’analogue du célèbre théorème de Markoff sur les nombres réels mal approximables.

Received: 2020-08-31
Revised: 2021-05-06
Accepted: 2021-06-23
Published online: 2023-05-12

DOI: 10.5802/aif.3522

Classification: 11J06, 11J70, 68R15
Keywords: Lagrange spectrum, Romik’s dynamical system, Diophantine approximation on a manifold
Mots-clés : Spectre de Lagrange, système dynamique de Romik, approximation diophantienne sur une variété

Author's affiliations:

Cha, Byungchul ¹; Kim, Dong Han ²

¹ Muhlenberg College 2400 Chew st Allentown, PA 18104 (USA)
² Department of Mathematics Education Dongguk University - Seoul 30 Pildong-ro 1-gil, Jung-gu Seoul 04620 (Korea)

License:

CC-BY-ND 4.0

Copyrights: The authors retain unrestricted copyrights and publishing rights

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     author = {Cha, Byungchul and Kim, Dong Han},
     title = {Intrinsic {Diophantine} approximation on the unit circle and its {Lagrange} spectrum},
     journal = {Annales de l'Institut Fourier},
     pages = {101--161},
     publisher = {Association des Annales de l{\textquoteright}institut Fourier},
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Cha, Byungchul; Kim, Dong Han. Intrinsic Diophantine approximation on the unit circle and its Lagrange spectrum. Annales de l'Institut Fourier, Volume 73 (2023) no. 1, pp. 101-161. doi : 10.5802/aif.3522. https://aif.centre-mersenne.org/articles/10.5802/aif.3522/

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