Harold S. Erazo

activity

20242026

collaborators

9 papers

math.NT2026

Lagrange spectrum for Diophantine approximations of complex numbers with real part equal to one half

Hao Cheng, Harold Erazo, Carlos Gustavo Moreira +1

Motivated by a theorem of A. Schmidt on the part of the complex Lagrange spectrum below $2$ , we study the restricted Lagrange spectrum $L_{\frac{1}{2} + i R}$ arising from th…

math.NT2026

The ratio spectrum of Lagrange constants under linear fractional transformations

Harold Erazo, Carlos Gustavo Moreira

In this note we solve a problem posed by Lagarias and Shallit concerning Lagrange constants under linear fractional transformations $M x = \frac{a x + b}{c x + d}$ . For an integer matrix $M…

math.NT2026

On very badly approximable numbers

Zhe Cao, Harold Erazo, Carlos Gustavo Moreira

We prove a refined version of Markov's theorem in Diophantine approximation. More precisely, we characterize completely the set of irrationals $x$ such that $\left|x-\frac{p}{q}\ri…

math.NT2026

On the geometry of the second Lagrange spectra

Hao Cheng, Harold Erazo, Carlos Gustavo Moreira +1

The Lagrange spectrum $L$ is the set of finite values of the best approximation constants $k (α) = lim sup_{∣ p ∣, ∣ q ∣ \to \infty} ∣ q (q α - p) ∣^{- 1}$ , where $α\in \mathbb{R}\setminus \mathbb{…

math.NT2025

Generalized Hausdorff dimension of irrationals with Lagrange value exactly 3

Carlos Gustavo Moreira, Harold Erazo, Nicolas Angelini

We study the generalized Hausdorff dimension of some natural subsets of $k^{- 1} (3)$ , where $k^{- 1} (3)$ consists of the real numbers $x$ for which $\left| x-\frac{p}{q} \right|<\fra…

math.NT2025

On irrationals with Lagrange value exactly 3

Zhe Cao, Harold Erazo, Carlos Gustavo Moreira

For $c > 0$ , let $X_{c}$ denote the set of $x \in R \ Q$ such that $x - \frac{p}{q} < \frac{1}{c q ^{2}}$ has only finitely many rational solutions $\frac…

math.NT2024

inf(M \ L)=3

Harold Erazo, Davi Lima, Carlos Matheus +2

The Lagrange and Markov spectra $L$ and $M$ describe the best constants of Diophantine approximations for irrational numbers and binary quadratic forms. In 1880, A. Markov showed t…

math.NT2024

On the classical Lagrange and Markov spectra: new results on the local dimension and the geometry of the difference set

Harold Erazo, Luke Jeffreys, Carlos Gustavo Moreira

Let $L$ and $M$ denote the classical Lagrange and Markov spectra, respectively. It is known that $L \subset M$ and that $M ∖ L \neq = \emptyset$ . Inspired by three questions as…

math.NT2024

New portions of $M ∖ L$ and a lower bound on the Hausdorff distance between $L$ and $M$

Clément Rieutord, Carlos Gustavo Moreira, Harold Erazo

Let $M$ and $L$ be the Markov and Lagrange spectra, respectively. It is known that $L$ is contained in $M$ and Freiman showed in 1968 that $M ∖ L \neq = \emptyset$ . In 2018 the…

Lagrange spectrum for Diophantine approximations of complex numbers with real part equal to one half

The ratio spectrum of Lagrange constants under linear fractional transformations

On very badly approximable numbers

On the geometry of the second Lagrange spectra

Generalized Hausdorff dimension of irrationals with Lagrange value exactly 3

On irrationals with Lagrange value exactly 3

inf(M \ L)=3

On the classical Lagrange and Markov spectra: new results on the local dimension and the geometry of the difference set

New portions of M∖L and a lower bound on the Hausdorff distance between L and M

New portions of $M ∖ L$ and a lower bound on the Hausdorff distance between $L$ and $M$