Let be the Lagrange spectrum arising from intrinsic Diophantine approximation on the unit circle by its rational points. We give a complete description of the structure of below its smallest accumulation point. To this end, we use digit expansions of points on , 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 is 2. Also we characterize the points on 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 le spectre de Lagrange associé à l’approximation diophantienne intrinsèque sur le cercle unité par ses points rationnels. Nous donnons une description complète de la structure de en dessous de son plus petit point d’accumulation. Pour cela, nous utilisons les développements infinis des points de 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 . Nous caractérisons également les développements de Romik des points de 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.
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Keywords: Lagrange spectrum, Romik’s dynamical system, Diophantine approximation on a manifold
Mot clés : Spectre de Lagrange, système dynamique de Romik, approximation diophantienne sur une variété

@article{AIF_2023__73_1_101_0, 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}, volume = {73}, number = {1}, year = {2023}, doi = {10.5802/aif.3522}, language = {en}, url = {https://aif.centre-mersenne.org/articles/10.5802/aif.3522/} }
TY - JOUR AU - Cha, Byungchul AU - Kim, Dong Han TI - Intrinsic Diophantine approximation on the unit circle and its Lagrange spectrum JO - Annales de l'Institut Fourier PY - 2023 SP - 101 EP - 161 VL - 73 IS - 1 PB - Association des Annales de l’institut Fourier UR - https://aif.centre-mersenne.org/articles/10.5802/aif.3522/ DO - 10.5802/aif.3522 LA - en ID - AIF_2023__73_1_101_0 ER -
%0 Journal Article %A Cha, Byungchul %A Kim, Dong Han %T Intrinsic Diophantine approximation on the unit circle and its Lagrange spectrum %J Annales de l'Institut Fourier %D 2023 %P 101-161 %V 73 %N 1 %I Association des Annales de l’institut Fourier %U https://aif.centre-mersenne.org/articles/10.5802/aif.3522/ %R 10.5802/aif.3522 %G en %F AIF_2023__73_1_101_0
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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