arXiv subjects

Carlos Gustavo Moreira

Publications and source records attributed to Carlos Gustavo Moreira.

At least 19 recordsLinked to original sources

arXiv2026

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

Motivated by a theorem of A. Schmidt on the part of the complex Lagrange spectrum below $2$, we study the restricted Lagrange spectrum $L_{\frac{1}{2}+i\mathbb{R}}$ arising from the approximation of complex numbers of the form $\frac{1}{2}+iα$, $α\in\mathbb{R}\setminus\mathbb{Q}$ by Gaussian rationals $p/q$ with $p,q\in\mathbb{Z}[i]$, $q\neq 0$. We show that this spectrum admits a description in terms of a dynamical spectrum associated with a real horseshoe. As a consequence, we obtain several fractal properties of $L_{\frac{1}{2}+i\mathbb{R}}$, such as continuity of the dimension function $t\mapsto\dim_H(L_{\frac{1}{2}+i\mathbb{R}}\cap(-\infty,t))$. We also prove that the set of complex numbers $z$ satisfying \begin{equation*} \left\lvert z-\frac{p}{q}\right\rvert\geq\frac{1}{2|q|^2}, \quad\text{for all } p,q\in\mathbb{Z}[i], q\neq 0, \end{equation*} is uncountable. In fact, we show that this inequality holds for every complex number of the form $z=\frac{1}{2}(1+iθ)$ where $θ\in\mathbb{R}\setminus\mathbb{Q}$ is a root of one of Schmidt's $C$-minimal forms.

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arXiv2026

Stability of compact actions and a result on divided differences

We study smooth locally free actions of ${\mathbb R}^n$ on manifolds $M$ of dimension $n+1$. We are interested in compact orbits and in compact actions: actions with all orbits compact. Given a compact orbit in a neighborhood of compact orbits, we give necessary and sufficient conditions for the existence of a $C^k$ perturbation with noncompact orbits in the given neighborhood. We prove that if such a perturbation exists it can be assumed to differ from the original action only in a smaller neighborhood of the initial orbit. As an application, for each $k$, we give examples of compact actions which admit $C^{k-1}$-perturbations with noncompact orbits but such that all $C^k$-perturbations are compact. The main result generalizes for $k > 1$ a previous result for the case $C^1$. A critical auxiliary result is an estimate on divided differences.

math.DS↗

arXiv2026

On $C^k$-functions mapping $\mathbb{Q}$ into itself and Mahler's problem on Liouville numbers

Liouville numbers form a classical class of transcendental real numbers characterized by exceptionally strong rational approximations. A theorem of Maillet shows that non-constant rational functions with rational coefficients preserve the Liouville property, motivating a question of Mahler on whether analogous phenomena hold for transcendental functions. In this paper, we address this problem for real functions of finite smoothness. For any $\varepsilon>0$, we construct an uncountable set of $C^k$-functions on $\mathbb{R}$, dense with respect to the topology of uniform convergence on compact sets, mapping $\mathbb{Q}$ into itself and satisfying $\operatorname{den}(f(p/q)) \le q^{2k+\varepsilon}$, and deduce that such functions preserve Liouville numbers. In contrast, we prove a rigidity result about a $C^{2k+1}$-function mapping $\mathbb{Q}$ into itself and satisfying $\operatorname{den}(f(p/q)) \ll q^k$.

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arXiv2026

The ratio spectrum of Lagrange constants under linear fractional transformations

In this note we solve a problem posed by Lagarias and Shallit concerning Lagrange constants under linear fractional transformations $Mx=\frac{ax+b}{cx+d}$. For an integer matrix $M$ with nonzero determinant and relatively prime entries, define the ratio spectrum \begin{equation*} \mathcal{V}(M)=\left\{\frac{k(Mx)}{k(x)}:x\in\mathrm{Bad}\right\}, \end{equation*} where $k(x)$ denotes the Lagrange constant of the irrational number $x$ and $\mathrm{Bad}$ is the set of badly approximable numbers. Lagarias and Shallit proved that \begin{equation*} \mathcal{V}(M)\subseteq\left[\frac{1}{|\det M|},|\det M|\right], \end{equation*} and asked for the determination of $\mathcal{V}(M)$. We prove that \begin{equation*} \mathcal{V}(M)=\left[\frac{1}{|\det M|},|\det M|\right]. \end{equation*}

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arXiv2026

Percolation on hierarchical lattices

We consider independent Bernoulli percolation on top of sequences of hierarchical graphs. Given a graph $G_{1}$ with two distinguished vertices $a_{1}$ and $b_{1}$, the hierarchical graph with seed $G_{1}$ is the sequence $\big( G_{k} \big)_{k \geq 1}$ resulting from the inductive procedure, where the graph $G_{k+1}$ is obtained from $G_{k}$ by replacing each of its edges with a copy of $G_{1}$, attached by the vertices $a_{1}$ and $b_{1}$. We prove that, under sharp hypotheses, percolation on these graphs presents a unique phase transition. Second, we establish the existence of several critical exponents in this context, such as the critical exponents for the correlation length $ν$, the surface tension $μ$, the one-arm exponent $α_{1}$. Several results are also obtained for their infinite counterpart $G_\infty$, which is the Benjamini-Schramm limit of $G_k$: uniqueness of the infinite cluster, continuity of $θ(p)$, existence of the percolation-probability exponent $β$ and scaling relations for the critical exponents $α_1$, $ν$ and $β$. Furthermore, we analyze noise sensitivity for crossing functions in $G_{k}$ and establish sharp noise sensitivity in this setting. Finally, we propose a setup where it is possible to verify the locality hypothesis, stating that the critical threshold for percolation is a local property, while critical exponents are determined by the global geometry of the graph. As a consequence of the techniques developed here, we also provide a necessary and sufficient condition for the existence of a unique fixed point for the map $p \mapsto \mathbb{E}_p[g]$ in $(0,1)$, where $g:\{0,1\}^n \to \{0,1\}$ is a nontrivial monotone Boolean function.

math.PR↗

arXiv2026

On the graph of the dimension function of the Lagrange and Markov spectra

We study the graph of the function $d(t)$ encoding the Hausdorff dimensions of the classical Lagrange and Markov spectra with half-infinite lines of the form $(-\infty, t)$. For this sake, we use the fact that the Hausdorff dimension of dynamically Cantor sets drop after erasing an element of its Markov partition to determine twelve nontrivial plateaux of $d(t)$. Next, we employ rigorous numerical methods (from our recent joint paper with Pollicott) to produce approximations of the graph of $d(t)$ between these twelve plateaux. As a corollary, we prove that the largest ten non-trivial plateaux of $d(t)$ are exactly those plateaux with lengths $> 0.005$.

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arXiv2026

Generalized Hausdorff dimension of irrationals with Lagrange value exactly 3

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 $\left| x-\frac{p}{q} \right|<\frac{1}{(3+\varepsilon)q^2}$ has infinitely many rational solutions $\frac{p}{q}$ for any $\varepsilon<0$ but only finitely many for any $\varepsilon>0$. It is well known that $k^{-1}(3)$ is an uncountable set with Hausdorff dimension zero. Given any dimension function $h$, we determine the exact "cut point" at which the generalized Hausdorff dimension $\mathcal{H}^h(k^{-1}(3))$ drops from infinity to zero. In particular we show that such a measure is always zero or not $σ$--finite, and, as an application, we can classify topologically $k^{-1}(3)$. Moreover, we show that the subset of attainable elements of $k^{-1}(3)$ has the same generalized Hausdorff dimension as $k^{-1}(3)$, but the subset of non--attainable elements of $k^{-1}(3)$ has a "strictly smaller" generalized Hausdorff dimension.

math.NT↗

arXiv2026

On very badly approximable numbers

We prove a refined version of Markov's theorem in Diophantine approximation. More precisely, we characterize completely the set of irrationals $x$ such that $\left|x-\frac{p}{q}\right|<\frac{1}{3q^2}$ has only finitely many rational solutions: their continued fraction is eventually a balanced sequence through a simple coding. As consequence, we show that all such numbers are either quadratic surds or transcendental numbers. In particular, for any algebraic real number $x$ of degree at least $3$ there are infinitely rational numbers $\frac{p}{q}$ such that $\left|x-\frac{p}{q}\right|<\frac{1}{3q^2}$.

math.NT↗

arXiv2026

On irrationals with Lagrange value exactly 3

For $c>0$, let $X_c$ denote the set of $x\in\mathbb{R}\backslash\mathbb{Q}$ such that $\left| x-\frac{p}{q} \right|<\frac{1}{cq^2}$ has only finitely many rational solutions $\frac{p}{q}$. It is a classical fact, known since the 1950s, that $X_c$ is uncountable for $c>3$ and countable for $c<3$. However, the cardinality of $X_3$ does not appear to be present in the literature. We prove that $X_3$ is uncountable. More generally, we show that for any $n\in\mathbb{N}\cup\{\infty\}$, the set of $x\in\mathbb{R}\backslash\mathbb{Q}$ with Lagrange value exactly $3$ and such that $\left| x-\frac{p}{q} \right|<\frac{1}{3q^2}$ has exactly $n$ rational solutions $\frac{p}{q}$ is also uncountable.

math.NT↗

arXiv2026

On the geometry of the second Lagrange spectra

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.

math.NT↗

arXiv2025

Continuity of fractal dimensions in conservative generic Markov and Lagrange dynamical spectra

Let $φ_0$ be a smooth conservative diffeomorphism of a compact surface $S$ and let $Λ_0$ be a transitive horseshoe of $φ_0$. Given a smooth real function $f$ defined in $S$ and a small smooth conservative perturbation $φ$ of $φ_0$, let $L_{φ, f}$ and $M_{φ, f}$ be respectively the Lagrange and Markov spectra associated to the hyperbolic continuation $Λ(φ)$ of the horseshoe $Λ_0$ and $f$. We show that for generic choices of $φ$ and $f$, the Hausdorff dimension of the sets $L_{φ, f}\cap (-\infty, t)$ and $M_{φ, f}\cap (-\infty, t)$ are equal and determine a continuous function as $t\in \mathbb{R}$ varies; generalizing then the Cerqueira-Matheus-Moreira theorem to horseshoes with arbitrary Hausdorff dimension.

math.DS↗

arXiv2025

On the complexity of subshifts and infinite words

We characterize the complexity functions of subshifts up to asymptotic equivalence. The complexity function of every aperiodic function is non-decreasing, submultiplicative and grows at least linearly. We prove that conversely, every function satisfying these conditions is asymptotically equivalent to the complexity function of a recurrent subshift, equivalently, a recurrent infinite word. Our construction is explicit, algorithmic in nature and is philosophically based on constructing certain 'Cantor sets of integers', whose 'gaps' correspond to blocks of zeros. We also prove that every non-decreasing submultiplicative function is asymptotically equivalent, up a linear error term, to the complexity function of a minimal subshift.

math.DS↗

arXiv2025

Complexity and recurrence in infinite words and related structures

We study the asymptotics and fine-scale behavior of quantitative combinatorial measures of infinite words and related dynamical and algebraic structures. We construct infinite recurrent words $w$ whose complexity functions $p_w(n)$ are arbitrarily close to linear, but whose discrete derivatives are not bounded from above by $p_w(n)/n$. Moreover, we construct words of polynomially bounded complexity whose discrete derivatives exceed $p_w(n)/n^\varepsilon$ infinitely often, for every given $\varepsilon>0$. These provide negative answers in a strong sense to an open question of Cassaigne from 1997, showing that his theorem on words of linear complexity is best possible. Next, we characterize, up to a linear multiplicative error, the complexity functions of strictly ergodic subshifts, showing that every non-decreasing, submultiplicative function arises in this setting. This gives the first `industrial' construction of strictly ergodic subshifts of prescribed subexponential complexity. We then investigate quantitative recurrence in uniformly recurrent words and, as an application, address a question of Bavula from 2006 related to holonomic inequalities on the spectrum of possible filter dimensions of simple associative algebras: we construct simple algebras of prescribed filter dimension in $[1,\infty)$ and essentially settling the problem entirely in the graded case. Throughout, we construct uniformly recurrent words of linear complexity and with arbitrary polynomial recurrence growth.

math.CO↗

arXiv2025

On fast Lyapunov spectra for Markov-Rényi maps

In this paper, we study the multifractal analysis for Markov-Rényi maps, which form a canonical class of piecewise differentiable interval maps, with countably many branches and may contain a parabolic fixed point simultaneously, and do not assume any distortion hypotheses. We develop a geometric approach, independent of thermodynamic formalism, to study the fast Lyapunov spectrum for Markov-Rényi maps. Our study can be regarded as a refinement of the Lyapunov spectrum at infinity. We demonstrate that the fast Lyapunov spectrum is a piecewise constant function, possibly exhibiting a discontinuity at infinity. Our results extend the works in \cite[Theorem 1.1]{FLWW13}, \cite[Theorem 1.2]{LR}, and \cite[Theorem 1.2]{FSW} from the Gauss map to arbitrary Markov-Rényi maps, and highlight several intrinsic differences between the fast Lyapunov spectrum and the classical Lyapunov spectrum. Moreover, we establish the upper and lower fast Lyapunov spectra for Markov-Rényi maps.

math.DS↗

arXiv2025

On the Exceptional Sets of Transcendental Analytic Functions in Several Variables with Integer Coefficients

In 2020, Marques and Moreira proved that every subset of $\overline{\mathbb{Q}} \cap B(0,1)$, which is closed under complex conjugation and contains $0$, is the exceptional set of uncountably many transcendental analytic functions with integer coefficients. In this paper, we extend this result to transcendental analytic functions in several variables. In particular, we show that every subset of $\overline{\mathbb{Q}}^m$ contained in the unit polydisc $Δ(0;1)$, closed under complex conjugation and containing the zero vector, is the exceptional set of uncountably many transcendental analytic functions in several variables with integer coefficients.

math.NT↗

arXiv2025

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

Let $L$ and $M$ denote the classical Lagrange and Markov spectra, respectively. It is known that $L\subset M$ and that $M\setminus L\neq\varnothing$. Inspired by three questions asked by the third author in previous work investigating the fractal geometric properties of the Lagrange and Markov spectra, we investigate the function $d_{loc}(t)$ that gives the local Hausdorff dimension at a point $t$ of $L'$. Specifically, we construct several intervals (having non-trivial intersection with $L'$) on which $d_{loc}$ is non-decreasing. We also prove that the respective intersections of $M'$ and $M''$ with these intervals coincide. Furthermore, we completely characterize the local dimension of both spectra when restricted to those intervals. Finally, we demonstrate the largest known elements of the difference set $M\setminus L$ and describe two new maximal gaps of $M$ nearby.

math.NT↗

arXiv2025

Hausdorff dimension of some subsets of the Lagrange and Markov spectra near $3$

We study the sets $\mathcal{L}$ and $\mathcal{M}\setminus\mathcal{L}$ near $3$, where $\mathcal{L}$ and $\mathcal{M}$ are the classical Lagrange and Markov spectra. More specifically, we construct a strictly decreasing sequence $\{a_r\}_{r\in \mathbb{N}}$ converging to $3$, such that for any $r$ one can find a subset $\mathcal{B}_r\subset (a_{r+1},a_r)\cap \mathcal{L}^{'}$ with the property that the Hausdorff dimension of $((a_{r+1},a_r)\cap \mathcal{L})\setminus \mathcal{B}_r$ is less than the Hausdorff dimension of $\mathcal{B}_r$ and for $t\in \mathcal{B}_r$ the sets of irrational numbers with Lagrange value bounded by $t$ and exactly $t$ respectively, have the same Hausdorff dimension. We also show that, as $t$ varies in $\mathcal{B}_r$, this Hausdorff dimension is a strictly increasing function. Finally, in relation to $\mathcal{M}\setminus \mathcal{L}$, we find $C>0$ such that we can bound from above the Hausdorff dimension of $(\mathcal{M}\setminus \mathcal{L})\cap (-\infty,3+ρ)$ by $\frac{\log (\abs{\log ρ})-\log (\log(\abs{\log ρ}))+C}{\abs{\log ρ}}$ if $ρ>0$ is small.

math.DS↗

arXiv2024

Stable intersections of Cantor sets and positive density of persistent tangencies for homoclinic bifurcations of automorphisms of $\mathbb{C}^2$

Let $\{f_μ\}_{μ\in \mathbb{D}}$ be a family of automorphisms of $\mathbb{C}^2$ unfolding a generic homoclinic tangency associated to a fixed point $p$ belonging to a horseshoe. We prove that if the linearized versions of the Cantor sets representing the local intersections of the stable and unstable manifolds of $p$ with the horseshoe have stable intersections, then the set of parameters $μ$ corresponding to automorphisms with persistent tangencies has positive density at $μ= 0$.

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