arXiv subjects

Vicente Saavedra-Araya

Publications and source records attributed to Vicente Saavedra-Araya.

5 recordsLinked to original sources

arXiv2026

Uniqueness of a topological Furstenberg system

Given a semigroup $G$ and a bounded function $f: G \to \mathbb{C}$, a topological Furstenberg system of $f$ is a topological dynamical system $\mathbb{X}=(X, (T_g)_{g \in G})$ that encodes the dynamical behaviour of $f$. We show that $\mathbb{X}$ is unique up to topological isomorphism, thus providing a topological analogue of the measurable case established by Bergelson and Ferré Moragues for amenable semigroups. We also provide necessary and sufficient conditions for subsets of a group to have isomorphic Furstenberg systems. In addition, we study sets with minimal Furstenberg systems and identify them as a special subclass of dynamically syndetic sets. Moreover, we use this notion to obtain a new characterization of sets of topological recurrence.

math.DS↗

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

Distribution of integers with digit restrictions via Markov chains

In this paper, we introduce a new technique to study the distribution in residue classes of sets of integers with digit and sum-of-digits restrictions. From our main theorem, we derive a necessary and sufficient condition for integers with missing digits to be uniformly distributed in arithmetic progressions, extending previous results going back to the work of Erdős, Mauduit and Sárközy. Our approach utilizes Markov chains and does not rely on Fourier analysis as many results of this nature do. Our results apply more generally to the class of multiplicatively invariant sets of integers. This class, defined by Glasscock, Moreira and Richter using symbolic dynamics, is an integer analogue to fractal sets and includes all missing digits sets. We address uniform distribution in this setting, partially answering an open question posed by the same authors.

math.DS↗

arXiv2025

Hitting times of shrinking targets: Transversality and an ergodic theorem

In this paper, we investigate ergodic and fractal properties of the sets $$Λ_y:=\Big\{n\in\mathbb{N}:\ \{u_ny\}\in I_n\Big\},$$ where $\{\cdot\}$ denotes the fractional part function, $(u_n)_{n\in\mathbb{N}}$ is an increasing sequence of real numbers, $y\in [0,1]$ and each $I_n$ is a finite union of intervals with decreasing Lebesgue measure. Our main result shows that, under suitable conditions, the set $Λ_y$ is good for pointwise convergence of ergodic averages for Lebesgue almost every $y\in [0,1]$. Furthermore, we prove a transversality phenomenon: for any fixed set $A\subseteq \mathbb{N}$, the sets $Λ_y$ and $A$ are geometrically independent for almost every $y\in[0,1]$, as witnessed by the integer-fractal dimension of their intersection

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↗