On the geometry of the second Lagrange spectra
Hao Cheng, Harold Erazo, Carlos Gustavo Moreira, Thiago Vasconcelos
TL;DR
The paper examines the geometry of the second Lagrange spectra, $L_2$ and $L_2^*$, through a dynamical lens and establishes a stark contrast between their dimension-theoretic properties. It derives explicit continued-fraction–based formulas for $k_2(\alpha)$ and $k_2^*(\alpha)$, then casts the spectra as dynamical Lagrange spectra for a Gauss-type map on a conservative horseshoe, enabling a continuity result for the dimension function $d_2(t)=HD(L_2\cap(-\infty,t))$ while proving a sharp discontinuity for $d_2^*(t)$ at $t=\tfrac{2}{3}$. The authors show that near $2/3$, $L_2^*$ exhibits rich structure driven by Gauss–Cantor constructions, with $d_2^*(\tfrac{2}{3})=0$ but $d_2^*(\tfrac{2}{3}+\varepsilon)=1$ for any $\varepsilon>0$, and provide a lower bound $HD(X(k))>\tfrac12$ for large $k$ to support these results. Overall, the work highlights how non-smoothmax-type observables and Cantor-set dynamics can produce discontinuities in dimension, contrasting with the smooth-case continuity for $L_2$.
Abstract
The Lagrange spectrum $L$ is the set of finite values of the best approximation constants $k(α)=\limsup_{|p|,|q|\to \infty}|q(qα-p)|^{-1}$, where $α\in \mathbb{R}\setminus \mathbb{Q}$. 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 $α$. Consequently, $k(α)=\limsup_{n\to\infty}|q_n(q_nα-p_n)|^{-1}$. 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 $α$ 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.