Approximation by non-convergents and second Lagrange spectrum

Gayfulin, Dmitry

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

Given an irrational number $\alpha$ consider its irrationality measure function

$${\psi }_{\alpha }\left(t\right)=\underset{1\le q\le t,q\in \mathbb{Z}}{min}\parallel q\alpha \parallel .$$

The set of all values of

$$\lambda \left(\alpha \right)=\underset{t\to \infty }{lim\; sup}\left({\left(t{\psi }_{\alpha }\left(t\right)\right)}^{-1}\right)$$

where $\alpha$ runs through the set $ℝ\setminus \mathbb{Q}$ is known as the Lagrange spectrum $𝕃$. It is very well studied. In the present paper, we consider another irrationality measure function ${\psi }_{\alpha }^{\left[2\right]}\left(t\right)$ which deals with rational approximations to $\alpha$ by non-convergents. Replacing the function ${\psi }_{\alpha }\left(t\right)$ in the definition of $𝕃$ by ${\psi }_{\alpha }^{\left[2\right]}\left(t\right)$, 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 ${𝕃}_2$.

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

$${\psi }_{\alpha }\left(t\right)=\underset{1\le q\le t,q\in \mathbb{Z}}{min}\parallel q\alpha \parallel .$$

L’ensemble $𝕃$ des valeurs de la fonction

$$\lambda \left(\alpha \right)=\underset{t\to \infty }{lim\; sup}\left({\left(t{\psi }_{\alpha }\left(t\right)\right)}^{-1}\right)$$

où $\alpha$ parcourt l’ensemble $ℝ\setminus \mathbb{Q},$ est appelé le spectre de Lagrange. Il est très bien étudié. Dans cet article, nous considérons une autre mesure d’irrationalité, ${\psi }_{\alpha }^{\left[2\right]}\left(t\right)$, qui traite l’approximation du nombre $\alpha$ par des rationnels non réduits. En remplaçant la fonction ${\psi }_{\alpha }$ par ${\psi }_{\alpha }^{\left[2\right]}$ 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 ${𝕃}_2$.

Classification : 11J06
Keywords: Continued fractions, Diophantine approximation

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