arXiv subjects

Rigoberto Zelada

Publications and source records attributed to Rigoberto Zelada.

10 recordsLinked to original sources

arXiv2026

Polynomial maps which are not good for nice recurrence and applications

Let $\mathbb F_2$ be the finite field with two elements and let $\mathbb F_2^ω$ denote the countably infinite-dimensional vector space over $\mathbb F_2$. We show that, unlike the case of polynomial maps $p:\mathbb Z\rightarrow\mathbb Z$ vanishing at zero which are always good for nice recurrence, there are polynomials $p:\mathbb F_2^ω\rightarrow \mathbb F_2^ω$ with $p(0_{\mathbb F_2^ω})=0_{\mathbb F_2^ω}$ which fail to have this property. This disproves a conjecture of Bergelson and McCutcheon (c. 2000), which predicted that for any countably infinite abelian groups $H$ and $G$, every polynomial map $p:H\to G$ with $p(0_H)=0_G$ is good for nice recurrence. Moreover, we develop a dynamical mechanism which shows that the magnitude of intersections along polynomial paths is degree-sensitive (even when one considers only weakly mixing systems). Among other things, we also show that the Furstenberg-Sarkozy theorem for $\mathbb F_2^ω$-valued polynomials of degree at most $d$ vanishing at zero is equivalent to a $d$-dimensional symmetric-difference weakening of the density polynomial Hales-Jewett conjecture. Thus, as we explain in detail in this paper, our observations not only shed new light on the phenomenon of polynomial recurrence but also constrain possible strategies for proving or disproving the density polynomial Hales-Jewett conjecture.

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arXiv2026

Coexistence of mixing and rigid behaviors in ergodic theory

In this paper we introduce and explore the notion of rigidity group, associated with a collection of finitely many sequences, and show that this concept has many, somewhat surprising characterizations of algebraic, spectral, and unitary nature. Furthermore, we demonstrate that these characterizations can be employed to obtain various results in the theory of generic Lebesgue-preserving automorphisms of $[0,1]$, IP-ergodic theory, multiple recurrence, additive combinatorics, and spectral theory. As a consequence of one of our results we show that given $(b_1,...b_\ell)\in\mathbb N^\ell$, there is no orthogonal vector $(a_1,\dots,a_\ell)\in\mathbb Z^\ell$ with some $|a_j|=1$ if and only if there is an increasing sequence of natural numbers $(n_k)_{k\in\mathbb N}$ with the property that for each $F\subseteq \{1,...,\ell\}$ there is a $μ$-preserving transformation $T_F:[0,1]\rightarrow[0,1]$ ($μ$ denotes the Lebesgue measure) such that for any measurable $A,B\subseteq [0,1]$, $$\lim_{k\rightarrow\infty}μ(A\cap T_F^{-b_jn_k}B)=\begin{cases} μ(A\cap B),\,\text{ if }j\in F,\\ μ(A)μ(B),\,\text{ if }j\not\in F. \end{cases}$$ We remark that this result has a natural extension to a wide class of families of sequences.

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arXiv2026

Sets of large values of polynomial multi-correlation functions

Let $p_1,...,p_L\in Z[x_1,...,x_d]$ be non-constant polynomials with zero constant term. The ergodic theoretical proofs of the polynomial and the IP-polynomial Szemeredi theorems as well as some of the ergodic-theoretical and combinatorial consequences of the Density Polynomial Hales-Jewett conjecture (DPHJ) naturally lead to the study of sets of large returns which are defined as $$ R_ε^{p_1,...,p_L}(A):=\{n\in Z^d\,|\,μ(A\cap T_1^{-p_1( n)}A\cap\cdots\cap T_L^{-p_L(n)}A)>μ^{L+1}(A)-ε\}, $$ where the $T_j$'s are commuting and invertible $μ$-preserving transformations, $A$ is measurable, and $ε>0$. We obtain new results dealing with the sets of the form $R_ε^{p_1,...,p_L}(A)$. Among other things, we show that every set of the form $R_ε^{p_1,...,p_L}(A)$ is syndetic if and only if $p_1,...,p_L$ are linearly independent, answering a question asked by Frantzikinakis-Kuca. Moreover, the linear independence of $p_1,...,p_L$ implies that every set of the form $R_ε^{p_1,...,p_L}(A)$ has the A-IP$^*$ property (="almost" IP$^*$ property), which is stronger than syndeticity. The following is one of the new combinatorial results obtained in this paper. Suppose that $p_1,...,p_L$ are linearly independent. For any set $E\subseteq Z^D$ with upper Banach density $d^*(E)>0$, any non-zero $v_1,..., v_L\in Z^D$, and any $ε>0$, the set $$ S_ε^{p_1,...,p_L}(E):=\{ n\in Z^d\,|\,d^*(E\cap (E-p_1(n)v_1)\cap \cdots\cap (E-p_L(n)v_L))>(d^*(E))^{L+1}-ε\} $$ is A-IP$^*$. Furthermore, we prove that when $D>L>1$, this result is sharp: the A-IP$^*$ property cannot be upgraded to IP$^*$. The techniques developed in this paper lead to some additional applications. For example, we show that an amplified form of the IP-polynomial Szemeredi theorem conjectured by Bergelson- McCutcheon follows from the DPHJ.

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arXiv2026

Multipliers and Disjointness from Mixing

In 2005, Parreau proved that if a measure preserving system is not strongly mixing then it contains a non-trivial factor that is disjoint from every strongly mixing system. Taking this construction as the starting point, we develop the complementary notions of $\mathcal U$-generated and $\mathcal U$-mixing systems, for a set $\mathcal U$ of ultrafilters, and use them to recover several classical results in ergodic theory as special cases of a unified framework. We prove that a system is $\mathcal U$-mixing if and only if it is disjoint from all $\mathcal U$-generated systems. In fact, we show that if $\mathcal Y$ is a $\mathcal U$-generated system and $\mathcal Z$ is disjoint from every $\mathcal U$-mixing system, then any joining of $\mathcal Y$ and $\mathcal Z$ remains disjoint from all $\mathcal U$-mixing systems. We also show that every partially rigid system is a finite extension of some $\mathcal{U}$-generated system.

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arXiv2026

Weak mixing for area preserving flows on surfaces

Let $(ϕ_t)$ be an area-preserving smooth flow on a compact, connected, orientable surface $\mathcal M$ with at least one but finitely many fixed points. Assume that $(ϕ_t)$ is analytic (up to a canonical change of coordinates) in the neighborhood of each saddle fixed point. We show that the flow $(ϕ_t)$ is weakly mixing on each of its (finitely many) quasi-minimal components.

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arXiv2024

Iterated differences sets, diophantine approximations and applications

Let $v$ be an odd real polynomial (i.e. a polynomial of the form $\sum_{j=1}^\ell a_jx^{2j-1}$). We utilize sets of iterated differences to establish new results about sets of the form $\mathcal R(v,ε)=\{n\in\mathbb{N}\,|\,\|v(n)\|{<ε\}}$ where $\|\cdot\|$ denotes the distance to the closest integer. We then apply the new diophantine results to obtain applications to ergodic theory and combinatorics. In particular, we obtain a new characterization of weakly mixing systems as well as a new variant of Furstenberg-Sárközy theorem.

math.CO↗

arXiv2024

Failure of Khintchine-type results along the polynomial image of IP$_0$ sets

In "IP-sets and polynomial recurrence", Bergelson, Furstenberg, and McCutcheon established the following far reaching extension of Khintchine's recurrence theorem: For any invertible probability preserving system $(X,\mathcal A,μ,T)$, any non-constant polynomial $p\in\mathbb Z[x]$ with $p(0)=0$, any $A\in\mathcal A$, and any $ε>0$, the set $$R_ε^p(A)=\{n\in\mathbb N\,|\,μ(A\cap T^{-p(n)}A)>μ^2(A)-ε\}$$ is IP$^*$, meaning that for any increasing sequence $(n_k)_{k\in\mathbb N}$ in $\mathbb N$, $$\text{FS}((n_k)_{k\in\mathbb N})\cap R_ε^p(A)\neq \emptyset,$$ where $$\text{FS}((n_k)_{k\in\mathbb N})=\{\sum_{j\in F}n_j\,|\,F\subseteq \mathbb N\,\text{ is finite}\text{ and }F\neq\emptyset\}=\{n_{k_1}+\cdots+n_{k_t}\,|\,k_1<\cdots 1$ and $p(0)=0$ there is an invertible probability preserving system $(X,\mathcal A,μ,T)$, a set $A\in\mathcal A$, and an $ε>0$ for which the set $R_ε^p(A)$ is not IP$_0^*$.

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arXiv2023

A non-mixing Arnold flow on a surface

We construct a smooth area preserving flow on a genus 2 surface with exactly one open uniquely ergodic component, that is asymmetrically bounded by separatrices of non-degenerate saddles and that is nevertheless not mixing.

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arXiv2022

Mixing and rigidity along asymptotically linearly independent sequences

We utilize Gaussian measure preserving systems to prove the existence and genericity of Lebesgue measure preserving transformations $T:[0,1]\rightarrow [0,1]$ which exhibit both mixing and rigidity behavior along families of asymptotically linearly independent sequences. Let $λ_1,...,λ_N\in[0,1]$ and let $ϕ_1,...,ϕ_N:\mathbb N\rightarrow\mathbb Z$ be asymptotically linearly independent (i.e. for any $(a_1,...,a_N)\in\mathbb Z^N\setminus\{\vec 0\}$, $\lim_{k\rightarrow\infty}|\sum_{j=1}^Na_jϕ_j(k)|=\infty$). Then the class of invertible Lebesgue measure preserving transformations $T:[0,1]\rightarrow[0,1]$ for which there exists a sequence $(n_k)_{k\in\mathbb N}$ in $\mathbb N$ with $$\lim_{k\rightarrow\infty}μ(A\cap T^{-ϕ_j(n_k) }B)= (1-λ_j)μ(A\cap B)+λ_jμ(A)μ(B),$$ for any measurable $A,B\subseteq [0,1]$ and any $j\in\{1,...,N\}$, is generic. This result is a refinement of a result due to A. M. Stëpin (see Theorem 2 in "Spectral properties of generic dynamical systems") and a generalization of a result due to V. Bergelson, S. Kasjan, and M. Lemańczyk (see Corollary F in "Polynomial actions of unitary operators and idempotent ultrafilters").

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arXiv2021

Strongly mixing systems are almost strongly mixing of all orders

We prove that any strongly mixing action of a countable abelian group on a probability space has higher order mixing properties. This is achieved via introducing and utilizing $\mathcal R$-limits, a notion of convergence which is based on the classical Ramsey Theorem. $\mathcal R$-limits are intrinsically connected with a new combinatorial notion of largeness which is similar to but has stronger properties than the classical notions of uniform density one and IP$^*$. While the main goal of this paper is to establish a $\textit{universal}$ property of strongly mixing actions of countable abelian groups, our results, when applied to $\mathbb Z$-actions, offer a new way of dealing with strongly mixing transformations. In particular, we obtain several new characterizations of strong mixing for $\mathbb Z$-actions, including a result which can be viewed as the analogue of the weak mixing of all orders property established by Furstenberg in the course of his proof of Szemerédi's theorem. We also demonstrate the versatility of $\mathcal R$-limits by obtaining new characterizations of higher order weak and mild mixing for actions of countable abelian groups.

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