arXiv subjects

Sebastián Donoso

Publications and source records attributed to Sebastián Donoso.

At least 19 recordsLinked to original sources

arXiv2026

Entropy and complexity in a strong orbit equivalence class via pseudo-Toeplitz subshifts

We develop a method for constructing minimal subshifts with a prescribed bound on the factor complexity within any strong orbit equivalence (SOE) class. This implies realizations of topological entropies within any SOE class, strengthening a series of classical results by Sugisaki. While prior work established complexity controls for zero-entropy systems, the approach of the present work extends to positive-entropy regimes. Our construction introduces the class of pseudo-Toeplitz subshifts, and we show that they exist in any SOE class. More precisely, for any $α\geq 1$ and sequence $g_n$ growing exponentially at rate $\log(α)$, we construct a system in this class whose topological entropy is $\log(α)$ and whose complexity grows strictly faster, or, under certain conditions, slower than $g_n$. As a consequence, any Choquet simplex can be realized as the set of invariant measures of a Toeplitz subshift with precisely controlled complexity, in either a zero or positive-entropy regime.

math.DS↗

arXiv2026

Non-determinism in group actions and topological minimal self-joinings

We study the topological and geometrical aspects of non-determinism in group actions, generalizing the notion of non-expansive direction for abelian actions. Under suitable conditions on the acting group, we establish restrictions on the collection of non-deterministic horoballs (or half-spaces) for any action. We illustrate our results in the case of free and nilpotent groups, expanding upon known cases for abelian groups. We then apply these findings to standing problems concerning minimal self-joinings in topological dynamics, which are of independent interest.

math.DS↗

arXiv2026

Pointwise ergodic averages along the Omega function in number fields

We study a general criterion for guaranteeing that an ergodic average exhibits the strong sweeping-out property. This result implies, in particular, the failure of pointwise convergence of these averages. Our result applies to averages along the Omega function in number fields, generalizing a result of Loyd. We also show that the averages \[\frac{1}{N^2}\sum_{1\leq m,n \leq N}f(T^{Ω(m^2+n^2)}x)\] exhibit the strong sweeping-out property, which answers affirmatively a question posed by Le, Moreira, Sun, and the second author. On the other hand, using number-theoretic methods, we establish the pointwise convergence of averages along the $Ω$ function defined on the ideals of a number field in uniquely ergodic systems. Using this dynamical framework, we also derive several natural number-theoretic consequences of independent interest.

math.DS↗

arXiv2026

Cube structures of the universal minimal system, nilsystems and applications

We propose and develop an approach to study nilsystems and their proximal extensions using cube structures associated with the universal minimal system. We provide alternative proofs for results regarding saturation properties of factor maps to maximal nilfactors in cubes, as well as new results and applications of independent interest to the structural theory of topological systems. In particular, we give a new proof that $\mathbf{RP}^{[d]}$ is an equivalence relation, building upon the distal case, by establishing a description of this relation in algebraic terms. This is new even for d=1.

math.DS↗

arXiv2026

Partition regularity in imaginary quadratic rings of integers

We obtain partition regularity results for homogeneous quadratic equations whose parametrized solutions admit nice factorizations into linear forms over rings of integers of imaginary quadratic fields. To do so, we develop number-theoretic results of independent interest on such fields, such as a characterization for aperiodic completely multiplicative functions, the Turán-Kubilius inequality, and a new concentration estimate for multiplicative functions.

math.CO↗

arXiv2026

Ergodic averages for commutative transformations along return times

In this paper, we extend recent results on the convergence of ergodic averages along sequences generated by return times to shrinking targets in rapidly mixing systems, partially answering questions posed by the first author, Maass and the third author. In particular, for a fixed parameter $a\in (0,1)$ and for generic $y\in [0,1]$, we establish both $L^2$ and pointwise convergence for single averages and multiple averages for commuting transformations along the sequences $(a_n(y))_{n\in \mathbb{N}}$, obtained by arranging the set $$\Big\{n\in\mathbb{N}: 0<2^ny \mod{1}<n^{-a} \Big\}$$ in an increasing order. We also obtain new results for semi-random ergodic averages along sequences of similar type.

math.DS↗

arXiv2025

Seminorm estimates and joint ergodicity for pairwise independent Hardy sequences

We develop a robust structure theory for multiple ergodic averages of commuting transformations along Hardy sequences of polynomial growth. We then apply it to derive a number of novel results on joint ergodicity, recurrence and convergence. In particular, we prove joint ergodicity for (a) pairwise independent Hardy sequences and weakly mixing transformations, (b) strongly independent Hardy sequences and ergodic transformations, (c) strongly irrationally independent Hardy sequences and totally ergodic transformations. We use these joint ergodicity results to provide new recurrence results for multidimensional patterns along strongly independent Hardy sequences, showing for instance that all subsets of $\mathbb{Z}^2$ of positive upper density contain patterns of the form $$ (m_1, m_2),\; (m_1 + \lfloor n^{\sqrt{2}}\rfloor, m_2),\; (m_1, m_2 + \lfloor n^{\sqrt{2}} + n^{1/2}\rfloor).$$ Last but not least, we positively resolve the joint ergodicity classification problem for pairwise independent Hardy sequences, of which the aforementioned families are special cases. While building on recent technical advances (e.g. PET coefficient tracking schemes and joint ergodicity criteria), our work introduces a number of technical developments of its own. We construct a suitable generalization of Host-Kra and box seminorms that quantitatively control ergodic averages along Hardy sequences. We subsequently use them to obtain Host-Kra seminorm estimates for averages along all pairwise independent Hardy sequences. Furthermore, we develop an ergodic version of the quantitative concatenation argument that has recently found extensive use in combinatorics, number theory and harmonic analysis. Lastly, we obtain new simultaneous Taylor approximations for Hardy sequences, a crucial ingredient to deal with the aforementioned classes of Hardy sequences.

math.DS↗

arXiv2025

A pointwise ergodic theorem along return times of rapidly mixing systems

We introduce a new class of sparse sequences that are ergodic and pointwise universally $L^2$-good for ergodic averages. That is, sequences along which the ergodic averages converge almost surely to the projection to invariant functions. These sequences are generated randomly as return or hitting times in systems exhibiting a rapid correlation decay. This can be seen as a natural variant of Bourgain's Return Times Theorem. As an example, we obtain that for any $a\in (0,1/2)$, the sequence $\left\{n\in\mathbb{N}:\ 2^ny\mod{1}\in (0,n^{-a})\right\}$ is ergodic and pointwise universally $L^2$-good for Lebesgue almost every $y\in [0,1]$. Our approach builds on techniques developed by Frantzikinakis, Lesigne, and Wierdl in their study of sequences generated by independent random variables, which we adapt to the non-independent case.

math.DS↗

arXiv2025

Resolving the joint ergodicity problem for Hardy sequences

The joint ergodicity classification problem aims to characterize those sequences which are jointly ergodic along an arbitrary dynamical system if and only if they satisfy two natural, simpler-to-verify conditions on this system. These two conditions, dubbed the difference and product ergodicity conditions, naturally arise from Berend and Bergelson's pioneering work on joint ergodicity. Elaborating on our earlier work, we investigate this problem for Hardy sequences of polynomial growth, this time without making any independence assumptions on the sequences. Our main result establishes the "difficult" direction of the problem: if a Hardy family satisfies the difference and product ergodicity conditions on a given system, then it is jointly ergodic for this system. We also find that, surprisingly, the converse fails for certain pathological families of Hardy sequences, even though it holds for all "reasonable" Hardy families. We conclude by suggesting potential fixes to the statement of this problem. New ideas of independent interest developed in this paper include the structure theory of a family of factors generalizing Host-Kra and box factors; a strengthening of Tao-Ziegler's concatenation results; and the most robust extension of a seminorm smoothing argument.

math.DS↗

arXiv2025

Asymptoticity, automorphism groups and strong orbit equivalence

Given any strong orbit equivalence class of minimal Cantor systems and any cardinal number that is finite, countable, or the continuum, we show that there exists a minimal subshift within the given class whose number of asymptotic components is exactly the given cardinal. For finite or countable ones, we explicitly construct such examples using $\mathcal{S}$-adic subshifts. We derived the uncountable case by showing that any topological dynamical system with countably many asymptotic components has zero topological entropy. We also construct systems with arbitrarily high subexponential word complexity with only one asymptotic class. We deduce that within any strong orbit equivalence class, there exists a subshift whose automorphism group is isomorphic to $\mathbb{Z}$.

math.DS↗

arXiv2025

On partial rigidity of $\mathcal{S}$-adic subshifts

We develop combinatorial tools to study partial rigidity within the class of minimal $\mathcal{S}$-adic subshifts. By leveraging the combinatorial data of well-chosen Kakutani-Rokhlin partitions, we establish a necessary and sufficient condition for partial rigidity. Additionally, we provide an explicit expression to compute the partial rigidity rate and an associated partial rigidity sequence. As applications, we compute the partial rigidity rate for a variety of constant length substitution subshifts, such as the Thue-Morse subshift, where we determine a partial rigidity rate of 2/3. We also exhibit non-rigid substitution subshifts with partial rigidity rates arbitrarily close to 1 and as a consequence, using products of the aforementioned substitutions, we obtain that any number in $[0, 1]$ is the partial rigidity rate of a system.

math.DS↗

arXiv2024

On recurrence for $\mathbb{Z}^d$-Weyl systems

We study the topological recurrence phenomenon of actions of locally compact abelian groups on compact metric spaces. In the case of $\mathbb{Z}^d$-actions we develop new techniques to analyze Bohr recurrence sets. These techniques include finding and manipulating correlations between the coordinates of the set of recurrence. Using this, we show that Bohr recurrence sets are recurrence sets for $\mathbb{Z}^d$-Weyl systems. This family encompasses, for example, all $\mathbb{Z}^d$-affine nilsystems. To our knowledge, this is the first result towards a positive answer to Katznelson question in the case of $\mathbb{Z}^d$-actions.

math.DS↗

arXiv2024

Joint transitivity for linear iterates

We establish sufficient and necessary conditions for the joint transitivity of linear iterates in a minimal topological dynamical system with commuting transformations. This result provides the first topological analogue of the classical Berend and Bergelson joint ergodicity criterion in measure-preserving systems.

math.DS↗

arXiv2024

Averages of completely multiplicative functions over the Gaussian integers -- a dynamical approach

We prove a pointwise convergence result for additive ergodic averages associated with certain multiplicative actions of the Gaussian integers. We derive several applications in dynamics and number theory, including: (i) Wirsing's theorem for Gaussian integers: if $f\colon \mathbb{G} \to \mathbb{R}$ is a bounded completely multiplicative function, then the following limit exists: $$\lim_{N \to \infty} \frac{1}{N^2} \sum_{1 \leq m, n \leq N} f(m + {\rm i} n).$$ (ii) An answer to a special case of a question of Frantzikinakis and Host: for any completely multiplicative real-valued function $f: \mathbb{N} \to \mathbb{R}$, the following limit exists: $$\lim_{N \to \infty} \frac{1}{N^2} \sum_{1 \leq m, n \leq N} f(m^2 + n^2).$$ (iii) A variant of a theorem of Bergelson and Richter on ergodic averages along the $Ω$ function: if $(X,T)$ is a uniquely ergodic system with unique invariant measure $μ$, then for any $x\in X$ and $f\in C(X)$, $$\lim_{N\to\infty}\frac{1}{N^2}\sum_{1 \leq m, n \leq N} f(T^{Ω(m^2 + n^2)}x)=\int_Xf \ dμ.$$

math.DS↗

arXiv2023

Joint ergodicity for functions of polynomial growth

We provide necessary and sufficient conditions for joint ergodicity results for systems of commuting measure preserving transformations for an iterated Hardy field function of polynomial growth. Our method builds on and improves recent techniques due to Frantzikinakis and Tsinas, who dealt with multiple ergodic averages along Hardy field functions; it also enhances an approach introduced by the authors and Ferré Moragues to study polynomial iterates. The more general expression, in which the iterate is a linear combination of a Hardy field function of polynomial growth and a tempered function, is studied as well.

math.DS↗

arXiv2023

Seminorms for multiple averages along polynomials and applications to joint ergodicity

Exploiting the recent work of Tao and Ziegler on a concatenation theorem on factors, we find explicit characteristic factors for multiple averages along polynomials on systems with commuting transformations, and use them to study criteria of joint ergodicity for sequences of the form $(T^{p_{1,j}(n)}_{1}\cdot\ldots\cdot T^{p_{d,j}(n)}_{d})_{n\in\mathbb{Z}},$ $1\leq j\leq k$, where $T_{1},\dots,T_{d}$ are commuting measure preserving transformations on a probability measure space and $p_{i,j}$ are integer polynomials. To be more precise, we provide a sufficient condition for such sequences to be jointly ergodic, giving also a characterization for sequences of the form $(T^{p(n)}_{i})_{n\in\mathbb{Z}}, 1\leq i\leq d$ to be jointly ergodic, answering a question due to Bergelson.

math.DS↗

arXiv2023

A geometric framework for asymptoticity and expansivity in topological dynamics

We develop a geometric framework to address asymptoticity and nonexpansivity in topological dynamics. Our framework can be applied when the acting group is second countable and locally compact. As an application, we show extensions of Schwartzman's theorem in this context. Also, we get new results when the acting groups is ${\mathbb Z}^d$: any half-space of ${\mathbb R}^d$ contains a vector defining a (oriented) nonexpansive direction in the sense of Boyle and Lind. Finally, we deduce rigidity properties of distal Cantor systems.

math.DS↗

arXiv2023

ALBETO and DistilBETO: Lightweight Spanish Language Models

In recent years there have been considerable advances in pre-trained language models, where non-English language versions have also been made available. Due to their increasing use, many lightweight versions of these models (with reduced parameters) have also been released to speed up training and inference times. However, versions of these lighter models (e.g., ALBERT, DistilBERT) for languages other than English are still scarce. In this paper we present ALBETO and DistilBETO, which are versions of ALBERT and DistilBERT pre-trained exclusively on Spanish corpora. We train several versions of ALBETO ranging from 5M to 223M parameters and one of DistilBETO with 67M parameters. We evaluate our models in the GLUES benchmark that includes various natural language understanding tasks in Spanish. The results show that our lightweight models achieve competitive results to those of BETO (Spanish-BERT) despite having fewer parameters. More specifically, our larger ALBETO model outperforms all other models on the MLDoc, PAWS-X, XNLI, MLQA, SQAC and XQuAD datasets. However, BETO remains unbeaten for POS and NER. As a further contribution, all models are publicly available to the community for future research.

cs.CL↗