Approximation by non-convergents and second Lagrange spectrum

Dmitry Gayfulin

doi:10.5802/jtnb.1306

Dmitry Gayfulin ^1,²

¹ Graz University of Technology, Institute of Analysis and Number Theory, Steyrergasse 30/II 8010 Graz (Austria)
² Big Data and Information Retrieval School Faculty of Computer Science National Research University Higher School of Economics 11 Pokrovsky boulevard Moscow 109028 (Russia)

Journal de théorie des nombres de Bordeaux, Tome 36 (2024) no. 3, pp. 1039-1051.

Résumé
Abstract

Etant donné un nombre irrationnel $α$ , on considère sa mesure d’irrationalité

ψ_{α} (t) = min_{1 \leq q \leq t, q \in ℤ} ∥ q α ∥ .

L’ensemble $𝕃$ des valeurs de la fonction

λ (α) = \underset{t \to \infty}{lim sup} ({(t ψ_{α} (t))}^{- 1})

où $α$ parcourt l’ensemble $ℝ ∖ ℚ,$ est appelé le spectre de Lagrange. Il est très bien étudié. Dans cet article, nous considérons une autre mesure d’irrationalité, $ψ_{α}^{[2]} (t)$ , qui traite l’approximation du nombre $α$ par des rationnels non réduits. En remplaçant la fonction $ψ_{α}$ par $ψ_{α}^{[2]}$ dans la définition de $𝕃$ , on obtient un ensemble $𝕃_{2}$ appelé le spectre de Lagrange d’ordre deux. Dans cet article, nous donnons la structure complète de la partie discrète initiale de $𝕃_{2}$ .

Given an irrational number $α$ consider its irrationality measure function

ψ_{α} (t) = min_{1 \leq q \leq t, q \in ℤ} ∥ q α ∥ .

The set of all values of

λ (α) = \underset{t \to \infty}{lim sup} ({(t ψ_{α} (t))}^{- 1})

where $α$ runs through the set $ℝ ∖ ℚ$ is known as the Lagrange spectrum $𝕃$ . It is very well studied. In the present paper, we consider another irrationality measure function $ψ_{α}^{[2]} (t)$ which deals with rational approximations to $α$ by non-convergents. Replacing the function $ψ_{α} (t)$ in the definition of $𝕃$ by $ψ_{α}^{[2]} (t)$ , we get a set $𝕃_{2}$ which is called the second Lagrange spectrum. In the present paper, we give the complete structure of the initial discrete part of $𝕃_{2}$ .

Reçu le : 2023-06-03
Révisé le : 2023-10-12
Accepté le : 2023-11-11
Publié le : 2025-03-31

DOI : 10.5802/jtnb.1306

Classification : 11J06
Mots-clés : Continued fractions, Diophantine approximation

Affiliations des auteurs :

Dmitry Gayfulin ^{1, 2}

¹ Graz University of Technology, Institute of Analysis and Number Theory, Steyrergasse 30/II 8010 Graz (Austria)
² Big Data and Information Retrieval School Faculty of Computer Science National Research University Higher School of Economics 11 Pokrovsky boulevard Moscow 109028 (Russia)

Licence :

CC-BY-ND 4.0

Droits d'auteur : Les auteurs conservent leurs droits

@article{JTNB_2024__36_3_1039_0,
     author = {Dmitry Gayfulin},
     title = {Approximation by non-convergents and second {Lagrange} spectrum},
     journal = {Journal de th\'eorie des nombres de Bordeaux},
     pages = {1039--1051},
     publisher = {Soci\'et\'e Arithm\'etique de Bordeaux},
     volume = {36},
     number = {3},
     year = {2024},
     doi = {10.5802/jtnb.1306},
     language = {en},
     url = {https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.1306/}
}

TY  - JOUR
AU  - Dmitry Gayfulin
TI  - Approximation by non-convergents and second Lagrange spectrum
JO  - Journal de théorie des nombres de Bordeaux
PY  - 2024
SP  - 1039
EP  - 1051
VL  - 36
IS  - 3
PB  - Société Arithmétique de Bordeaux
UR  - https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.1306/
DO  - 10.5802/jtnb.1306
LA  - en
ID  - JTNB_2024__36_3_1039_0
ER  -

%0 Journal Article
%A Dmitry Gayfulin
%T Approximation by non-convergents and second Lagrange spectrum
%J Journal de théorie des nombres de Bordeaux
%D 2024
%P 1039-1051
%V 36
%N 3
%I Société Arithmétique de Bordeaux
%U https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.1306/
%R 10.5802/jtnb.1306
%G en
%F JTNB_2024__36_3_1039_0

Dmitry Gayfulin. Approximation by non-convergents and second Lagrange spectrum. Journal de théorie des nombres de Bordeaux, Tome 36 (2024) no. 3, pp. 1039-1051. doi : 10.5802/jtnb.1306. https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.1306/

Bibliographie
Cité par

[1] Thomas W. Cusick; Mary E. Flahive The Markoff and Lagrange Spectra, Mathematical Surveys and Monographs, 30, American Mathematical Society, 1989, ix+97 pages | DOI | MR | Zbl

[2] Jacques Lesca Sur les approximations diophantiennes à une dimension, Ph. D. Thesis, Université de Grenoble (France) (1968) (Doctorat d’État)

[3] Davi Lima; Carlos Matheus; Carlos Gustavo Moreira; Sandoel Vieira $𝕄 ∖ 𝕃$ near $3$ , Mosc. Math. J., Volume 21 (2021) no. 4, pp. 767-788 | DOI | MR | Zbl

[4] Andrei A. Markoff Sur les formes quadratiques binaires indéfinies, Math. Ann., Volume 15 (1879), pp. 381-406 | DOI | MR | Zbl

[5] Andrei A. Markoff Sur les formes quadratiques binaires indéfinies. II, Math. Ann., Volume 17 (1880), pp. 379-399 | DOI | Zbl

[6] Carlos Matheus; Carlos Gustavo Moreira Fractal geometry of the complement of Lagrange spectrum in Markov spectrum, Comment. Math. Helv., Volume 95 (2020) no. 3, pp. 593-633 | DOI | MR | Zbl

[7] Carlos Matheus; Carlos Gustavo Moreira Diophantine approximation, Lagrange and Markov spectra, and dynamical Cantor sets, Notices Am. Math. Soc., Volume 68 (2021) no. 8, pp. 1301-1311 | MR | Zbl

[8] Seigo Morimoto Zur Theorie der Approximation einer irrationalen Zahl durch rationale Zahlen, Tôhoku Math. J., Volume 45 (1938), pp. 177-187 | Zbl

[9] Nikolay G. Moshchevitin Über die Funktionen des Irrationalitatsmaßes, Analytic and probabilistic methods in number theory. Proceedings of the sixth international conference, Palanga, Lithuania, September 11–17, 2016, Vilnius University Publishing House (2017), pp. 123-148 | MR | Zbl

[10] Andrew M. Rockett; Peter Szüsz Continued fractions, World Scientific, 1992, ix+188 pages | DOI | MR | Zbl

[11] Pavel Semenyuk On a problem related to “second” best approximations to a real number, Mosc. J. Comb. Number Theory, Volume 12 (2023) no. 2, pp. 175-180 | DOI | MR | Zbl

Cité par Sources :