Intrinsic Diophantine approximation on the unit circle and its Lagrange spectrum
Annales de l'Institut Fourier, Online first, 61 p.

Let (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 (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 (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 (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 (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 (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:
Revised:
Accepted:
Online First:
DOI: 10.5802/aif.3522
Classification: 11J06,  11J70,  68R15
Keywords: Lagrange spectrum, Romik’s dynamical system, Diophantine approximation on a manifold
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)
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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, Online first, 61 p.

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