On the geometry of the second Lagrange spectra

TL;DR

This paper investigates the geometric properties of second Lagrange spectra, proving the continuity of one associated Hausdorff dimension function and the discontinuity of another, revealing distinct fractal structures.

Contribution

It introduces and analyzes two variants of second Lagrange spectra, establishing their Hausdorff dimension functions' continuity and discontinuity respectively, and clarifies their geometric differences.

Findings

The function d_2(t) is continuous.

The function d_2^*(t) is discontinuous.

d_2^*(t) takes only values 0 and 1.

Abstract

The Lagrange spectrum $L$ is the set of finite values of the best approximation constants $k(\alpha)=\limsup_{|p|,|q|\to \infty}|q(q\alpha-p)|^{-1}$, where $\alpha\in \mathbb{R}\setminus \mathbb{Q}$. It is a classical result that the pairs $(p,q)$ attaining these approximation constants arise from the convergents $(p_n,q_n)$ of the continued fraction of $\alpha$. Consequently, $k(\alpha)=\limsup_{n\to\infty}|q_n(q_n\alpha-p_n)|^{-1}$. Moreira proved that the function $d(t)=HD(L\cap(-\infty,t))$ where $HD$ denotes Hausdorff dimension, is continuous. Second Lagrange spectra are defined analogously to the classical Lagrange spectrum, but are associated with the problem of approximating an irrational number $\alpha$ by rational numbers $\frac{p}{q}$ that are not convergents of its continued fraction expansion. Two natural definitions arise depending on whether rational multiples $(p,q)=(kp_n,kq_n),k\geq 2$ which represent the same rational numbers as convergents, are allowed or excluded. Based on this distinction, Moshchevitin introduced two second Lagrange spectra, denoted $L_2$ and $L_2^*$. We prove that the function $d_2(t)=HD(L_2\cap (-\infty,t))$ is continuous, whereas $d_2^*(t)=HD(L_2^*\cap (-\infty,t))$ is discontinuous and assumes only the values 0 and 1.