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
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
Mot 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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     year = {2023},
     doi = {10.5802/aif.3522},
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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/

References
Cited by

[1] Abe, Ryuji; Aitchison, Iain R. Geometry and Markoff’s spectrum for $ℚ (i)$ , I, Trans. Amer. Math. Soc., Volume 365 (2013) no. 11, pp. 6065-6102 https://mathscinet-ams-org.muhlenberg.idm.oclc.org/mathscinet-getitem?mr=3091276 | DOI | MR | Zbl

[2] Abe, Ryuji; Aitchison, Iain R.; Rittaud, Benoît Two-color Markoff graph and minimal forms, Int. J. Number Theory, Volume 12 (2016) no. 4, pp. 1093-1122 https://mathscinet-ams-org.muhlenberg.idm.oclc.org/mathscinet-getitem?mr=3484300 | DOI | MR | Zbl

[3] Abe, Ryuji; Rittaud, Benoît On palindromes with three or four letters associated to the Markoff spectrum, Discrete Math., Volume 340 (2017) no. 12, pp. 3032-3043 https://mathscinet-ams-org.muhlenberg.idm.oclc.org/mathscinet-getitem?mr=3698093 | DOI | MR | Zbl

[4] Aigner, Martin Markov’s theorem and 100 years of the uniqueness conjecture, Springer, Cham, 2013, x+257 pages https://mathscinet-ams-org.muhlenberg.idm.oclc.org/mathscinet-getitem?mr=3098784 (A mathematical journey from irrational numbers to perfect matchings) | DOI | MR | Zbl

[5] Barning, F. J. M. On Pythagorean and quasi-Pythagorean triangles and a generation process with the help of unimodular matrices, Math. Centrum Amsterdam Afd. Zuivere Wisk., Volume 1963 (1963) no. ZW-011, p. 37 https://mathscinet-ams-org.muhlenberg.idm.oclc.org/mathscinet-getitem?mr=190077 | MR | Zbl

[6] Berggren, B. Pytagoreiska triangular, Tidskr. Elementär Mat. Fys. Kemi, Volume 17 (1934), pp. 129-139 | Zbl

[7] Berstel, Jean; Lauve, Aaron; Reutenauer, Christophe; Saliola, Franco V. Combinatorics on words. Christoffel words and repetitions in words, CRM Monograph Series, 27, American Mathematical Society, Providence, RI, 2009, xii+147 pages https://mathscinet-ams-org.muhlenberg.idm.oclc.org/mathscinet-getitem?mr=2464862 | DOI | MR | Zbl

[8] Boca, Florin P.; Merriman, Claire Coding of geodesics on some modular surfaces and applications to odd and even continued fractions, Indag. Math. (N.S.), Volume 29 (2018) no. 5, pp. 1214-1234 https://mathscinet-ams-org.muhlenberg.idm.oclc.org/mathscinet-getitem?mr=3853422 | DOI | MR | Zbl

[9] Bombieri, Enrico Continued fractions and the Markoff tree, Expo. Math., Volume 25 (2007) no. 3, pp. 187-213 https://mathscinet-ams-org.muhlenberg.idm.oclc.org/mathscinet-getitem?mr=2345177 | DOI | MR | Zbl

[10] Borel, J.-P.; Laubie, F. Quelques mots sur la droite projective réelle, J. Théor. Nombres Bordeaux, Volume 5 (1993) no. 1, pp. 23-51 https://mathscinet-ams-org.muhlenberg.idm.oclc.org/mathscinet-getitem?mr=1251226 | DOI | Numdam | MR | Zbl

[11] Cha, Byungchul; Chapman, Heather; Gelb, Brittany; Weiss, Chooka Lagrange spectrum of a circle over the Eisensteinian field, Monatsh. Math., Volume 197 (2022) no. 1, pp. 1-55 | DOI | MR | Zbl

[12] Cha, Byungchul; Kim, Dong Han Number theoretical properties of Romik’s dynamical system, Bull. Korean Math. Soc., Volume 57 (2020) no. 1, pp. 251-274 https://mathscinet-ams-org.muhlenberg.idm.oclc.org/mathscinet-getitem?mr=4060196 | DOI | MR | Zbl

[13] Cha, Byungchul; Nguyen, Emily; Tauber, Brandon Quadratic forms and their Berggren trees, J. Number Theory, Volume 185 (2018), pp. 218-256 https://mathscinet-ams-org.muhlenberg.idm.oclc.org/mathscinet-getitem?mr=3734349 | DOI | MR | Zbl

[14] Cohn, Harvey Approach to Markoff’s minimal forms through modular functions, Ann. of Math. (2), Volume 61 (1955), pp. 1-12 https://mathscinet-ams-org.muhlenberg.idm.oclc.org/mathscinet-getitem?mr=67935 | DOI | MR | Zbl

[15] Cusick, Thomas W.; Flahive, Mary E. The Markoff and Lagrange spectra, Mathematical Surveys and Monographs, 30, American Mathematical Society, Providence, RI, 1989, x+97 pages https://mathscinet-ams-org.muhlenberg.idm.oclc.org/mathscinet-getitem?mr=1010419 | DOI | MR | Zbl

[16] Fishman, Lior; Kleinbock, Dmitry; Merrill, Keith; Simmons, David Intrinsic Diophantine approximation on manifolds: general theory, Trans. Amer. Math. Soc., Volume 370 (2018) no. 1, pp. 577-599 https://mathscinet-ams-org.muhlenberg.idm.oclc.org/mathscinet-getitem?mr=3717990 | DOI | MR | Zbl

[17] Kim, Dong Han; Lee, Seul Bee; Liao, Lingmin Odd-odd continued fraction algorithm, Monatsh. Math., Volume 198 (2022) no. 2, pp. 323-344 | DOI | MR | Zbl

[18] Kleinbock, Dmitry; Merrill, Keith Rational approximation on spheres, Israel J. Math., Volume 209 (2015) no. 1, pp. 293-322 https://mathscinet-ams-org.muhlenberg.idm.oclc.org/mathscinet-getitem?mr=3430242 | DOI | MR | Zbl

[19] Kopetzky, Hans Günther Rationale Approximationen am Einheitskreis, Monatsh. Math., Volume 89 (1980) no. 4, pp. 293-300 https://mathscinet-ams-org.muhlenberg.idm.oclc.org/mathscinet-getitem?mr=587047 | DOI | MR | Zbl

[20] Kopetzky, Hans Günther Über das Approximationsspektrum des Einheitskreises, Monatsh. Math., Volume 100 (1985) no. 3, pp. 211-213 https://mathscinet-ams-org.muhlenberg.idm.oclc.org/mathscinet-getitem?mr=812612 | DOI | MR | Zbl

[21] Malyšev, A. V. Markov and Lagrange spectra (a survey of the literature), Zap. Naučn. Sem. Leningrad. Otdel. Mat. Inst. Steklov. (LOMI), Volume 67 (1977), p. 5-38, 225 https://mathscinet-ams-org.muhlenberg.idm.oclc.org/mathscinet-getitem?mr=0441876 Studies in number theory (LOMI), 4 | MR

[22] Markoff, A. Sur les formes quadratiques binaires indéfinies, Math. Ann., Volume 15 (1879), pp. 381-409 | DOI | Zbl

[23] Markoff, A. Sur les formes quadratiques binaires indéfinies. II, Math. Ann., Volume 17 (1880) no. 3, pp. 379-399 | DOI | Zbl

[24] Moshchevitin, Nikolay Über die rationalen Punkte auf der Sphäre, Monatsh. Math., Volume 179 (2016) no. 1, pp. 105-112 https://mathscinet-ams-org.muhlenberg.idm.oclc.org/mathscinet-getitem?mr=3439274 | DOI | MR | Zbl

[25] Panti, Giovanni Billiards on Pythagorean triples and their Minkowski functions, Discrete Contin. Dyn. Syst., Volume 40 (2020) no. 7, pp. 4341-4378 https://mathscinet-ams-org.muhlenberg.idm.oclc.org/mathscinet-getitem?mr=4097546 | DOI | MR | Zbl

[26] Reutenauer, Christophe Mots de Lyndon généralisés, Sém. Lothar. Combin., Volume 54 (2005/07), B54h, 16 pages https://mathscinet-ams-org.muhlenberg.idm.oclc.org/mathscinet-getitem?mr=2223030 | MR | Zbl

[27] Reutenauer, Christophe Christoffel words and Markoff triples, Integers, Volume 9 (2009), pp. A26, 327-332 https://mathscinet-ams-org.muhlenberg.idm.oclc.org/mathscinet-getitem?mr=2534916 | DOI | MR | Zbl

[28] Romik, Dan The dynamics of Pythagorean triples, Trans. Amer. Math. Soc., Volume 360 (2008) no. 11, pp. 6045-6064 https://mathscinet-ams-org.muhlenberg.idm.oclc.org/mathscinet-getitem?mr=2425702 | DOI | MR | Zbl

[29] Schmidt, Asmus L. Diophantine approximation of complex numbers, Acta Math., Volume 134 (1975), pp. 1-85 https://mathscinet-ams-org.muhlenberg.idm.oclc.org/mathscinet-getitem?mr=422168 | DOI | MR | Zbl

[30] Schmidt, Asmus L. On $C$ -minimal forms, Math. Ann., Volume 215 (1975), pp. 203-214 https://mathscinet-ams-org.muhlenberg.idm.oclc.org/mathscinet-getitem?mr=376530 | DOI | MR | Zbl

[31] Series, Caroline The modular surface and continued fractions, J. London Math. Soc. (2), Volume 31 (1985) no. 1, pp. 69-80 https://mathscinet-ams-org.muhlenberg.idm.oclc.org/mathscinet-getitem?mr=810563 | DOI | MR | Zbl

[32] Short, Ian; Walker, Mairi Even-integer continued fractions and the Farey tree, Symmetries in graphs, maps, and polytopes (Springer Proc. Math. Stat.), Volume 159, Springer, 2016, pp. 287-300 https://mathscinet-ams-org.muhlenberg.idm.oclc.org/mathscinet-getitem?mr=3516227 | DOI | MR | Zbl

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