arXiv subjects

Chunlin Liu

Publications and source records attributed to Chunlin Liu.

At least 19 recordsLinked to original sources

arXiv2026

A Uniform Product-Difference Theorem for Dense Subsets of $\mathbb Z^2$

We establish a uniform product-difference theorem for dense subsets of $\mathbb Z^2$, which gives an affirmative answer to Problem~2 of Fish and, as consequences, to both parts of his Problem~1. More precisely, we prove that for every $\delta>0$ there exists an integer $K(\delta)\geq 1$ such that every set $E\subseteq\mathbb Z^2$ with upper Banach density $d^\star(E)\geq\delta$ satisfies \[ K(\delta)\mathbb Z \subseteq \{ab:(a,b)\in E-E\}. \] As consequences, we obtain affirmative answers to both parts of Fish's Problem~1: for positive-density sets $E_1,E_2\subseteq\mathbb Z$ and $E\subseteq\mathbb Z^2$, respectively, the sets \[ (E_1-E_1)^2-(E_2-E_2)^2 \quad\text{and}\quad \{x^2-y^2:(x,y)\in E-E\} \] contain nontrivial ideals of $\mathbb Z$, with generators depending only on the corresponding density thresholds. In particular, the latter result also settles a conjecture of Davies concerning differences of the indefinite quadratic form $x^2-y^2$ in dense subsets of $\mathbb Z^2$.

math.CO↗

arXiv2026

Joint Point-Distality and structure theory for commuting actions

Let $G$ and $H$ act commutatively and minimally on a compact metrizable space $X$. We prove that if the two subactions are point-distal, equivalently HPI, then the action generated by them is again point-distal. We also construct a common highly proximal extension on which both subactions are strictly HPI. As consequences, transitivity of mixed products and linear iterates upgrades to minimality in the point-distal category. We then compare the canonical structure of the two subactions with that of the joint action. The maximal highly proximal operations coincide, whereas the maximal joint isometric factor over a common factor is the meet of the two subaction-isometric factors. This gives a recursive description of the joint Furstenberg tower. Finally, we construct commuting minimal distal homeomorphisms of $\mathbb T^3$ whose canonical Furstenberg towers are different, showing that the meet formula can be strict.

math.DS↗

arXiv2026

Does Forgetting Transfer Across Modalities? A Real-World Benchmark for Cross-Modal Knowledge Unlearning Evaluation

Vision-Language Models (VLMs), like Large Language Models (LLMs), may memorize sensitive, copyrighted, or harmful knowledge from their pretraining corpora. Removing such knowledge is essential for building trustworthy AI systems. However, existing studies primarily focus on forgetting within individual modalities. Although recent work has begun to explore cross-modal consistency in unlearning, the cross-modal transfer of real-world knowledge unlearning remains insufficiently studied. To address this gap, we introduce UNLINK-VL, a real-world benchmark for cross-modal knowledge unlearning in VLMs. Under a post-hoc unlearning setting in which the original forget and retain corpora are unavailable, UNLINK-VL selects visually identifiable real-world entities as unlearning targets and associates them with corresponding images and one-hop and multi-hop facts derived from Wikidata. The benchmark comprises four complementary subsets that evaluate direct forgetting of target knowledge, the propagation of forgetting through relational knowledge, the preservation of related non-target knowledge, and robustness to semantically equivalent queries. We train models under text-only and multimodal unlearning settings and evaluate forgetting effectiveness and retained utility across textual, visual, and cross-modal scenarios. Extensive experiments reveal a pronounced asymmetry in cross-modal transfer: multimodal unlearning remains effective under textual evaluation, whereas text-only unlearning transfers poorly to visual and cross-modal scenarios. Meanwhile, the evaluated methods largely preserve the models' general capabilities. These findings demonstrate that relying solely on intra-modal evaluation, particularly text-only evaluation, may substantially overestimate the effectiveness of knowledge unlearning in VLMs, underscoring the need for cross-modal unlearning and evaluation.

cs.AI↗

arXiv2026

Conditional Topomorphic Degree and Multivariate Mean Equicontinuity for Minimal Amenable Group Actions

Let $G$ be a countably infinite discrete amenable group acting minimally on a compact metric space $X$, and let $\pi:X\to X_{\mathrm{eq}}$ be the maximal equicontinuous factor map. We introduce the \emph{conditional topomorphic degree} $d:=\tdeg_G(X)\in\mathbb N\cup\{\infty\}$, which, when finite, is the least integer such that $\pi$ is an at most $d$-to-one topomorphic extension. We prove that, for every $r\ge2$, Weyl mean $r$-equicontinuity, mean $r$-equicontinuity along some F{\o}lner sequence, and $d\le r-1$ are equivalent. For minimal $\mathbb Z$-systems, this settles a conjecture of Breitenb\"ucher, Haupt, and J\"ager. We further establish the decomposition formula $d=\sum_{\mu\in\mathcal M_G^e(X)}\iota_\mu\exp\bigl(h_\mu^*(G)\bigr),$ where $\iota_\mu$ is the degree of the factor map from the measure-theoretic maximal compact factor associated with $\mu$ onto $X_{\mathrm{eq}}$, and $h_\mu^*(G)$ is the maximal measure sequence entropy of $\mu$. As further consequences of the decomposition formula, we show that for every finite $N$ with $2\le N\le d$, the system admits an essential IT $N$-tuple. Consequently, $h_{\mathrm{top}}^*(X,G)\ge \log d.$ This strengthens a lower bound of Liu, Wang, and Xu by also detecting compact multiplicities. As an application, we answer a question of G\'omez, Le\'on-Torres, and Mu\~noz-L\'opez. If $G$ contains a finite-index normal subgroup isomorphic to $\mathbb Z^r$, then, for every $m\ge2$, there exists a free minimal uniquely ergodic zero-entropy finite-alphabet $G$-subshift with maximal topological sequence entropy $\log m$, an essential IT $m$-tuple, and no essential IN $(m+1)$-tuple. For $G=\mathbb Z$, the alphabet may be chosen to have exactly $m$ symbols. Finally, we realize every finite multiplicity profile by a zero-entropy minimal almost one-to-one extension of an irrational circle rotation.

math.DS↗

arXiv2026

Li--Yorke Chaos Along Any Infinite Sequence: Relative Mixing, Sofic and Rokhlin Entropy

Let $G$ be a countably infinite discrete group and let $\pi:(X,\mu,G)\to(Y,\nu,G)$ be a nontrivial relatively mixing extension, where $X$ is a compact metrizable $G$-space. We prove that there exists a constant $\delta>0$ such that, for every injective sequence $(s_i)_{i\geq 1}$ in $G$, there is a Cantor set $K_{(s_i)}\subseteq X$ whose distinct points $x,x'$ satisfy \[ \liminf_{i\to\infty}\rho(s_i x,s_i x')=0, \qquad \limsup_{i\to\infty}\rho(s_i x,s_i x')>\delta. \] The method also yields higher-order scrambled Cantor sets. As a principal application, for a sofic group $G$, positive topological sofic entropy implies the preceding conclusion, answering a question of Huang, Li, and Ye. The same conclusion also holds for actions of arbitrary countably infinite discrete groups admitting an essentially free invariant measure of positive Rokhlin entropy.

math.DS↗

arXiv2026

Mean Equicontinuity and Related Properties in Hyperspace and Measure Dynamics

For a dynamical system $(X,T)$ we consider the induced dynamical systems $(\myper(X),T)$ and $(\hyper(X),T)$, consisting of Borel probability measures and closed non-empty subsets, respectively. We show that diam-mean equicontinuity of $(X,T)$ is equivalent to the diam-mean equicontinuity of $(\myper(X),T)$. Furthermore, we establish that $(X,T)$ is mean equicontinuous, iff $(\myper(X),T)$ is mean equicontinuous, iff $(\myper(X),T)$ is weakly-mean equicontinuous. For $(\hyper(X),T)$ the situation is different. It is not hard to see that the diam-mean equicontinuity of $(\hyper(X),T)$ implies the diam-mean equicontinuity of $(X,T)$. We provide examples for which $(X,T)$ is diam-mean equicontinuous, while $(\hyper(X),T)$ is not diam-mean equicontinuous. We prove that $(\hyper(X),T)$ is diam-mean equicontinuous, iff $(\hyper(X),T)$ is mean equicontinuous, iff $(\hyper(X),T)$ is weakly-mean equicontinuous. We present our results in the context of continuous surjective maps $T\colon X\to X$ and discuss why they also hold for actions of locally compact $\sigma$-compact amenable groups.

math.DS↗

arXiv2026

Dynamical Cantor Staircase Functions and The Small Flow Boundary Property

The small flow boundary property (SFBP), introduced by Burguet for fixed-point free topological flows, is a non-trivial generalization of the small boundary property (SBP). We characterize when a time-discretization of such a flow satisfies the SBP and deduce that an SFBP flow admitting an aperiodic time-discretization has vanishing mean dimension. Furthermore, we introduce a new quantity, \textit{flow-generated entropy}, for a factor between a flow and a time-discretization, quantifying the dynamical complexity inherited from the flow itself. This is used in order to establish that any time-discretization of a flow with SFBP admits factors of arbitrarily small flow-generated entropy separating any fixed pair of distinct points. The argument relies on a construction of a dynamical version of the Cantor staircase function. Finally, the appendix includes proofs of fundamental properties of the marker property which have not yet appeared in the literature.

math.DS↗

arXiv2026

The Bishop--Phelps--Bollob\'as Property for Extremally Disconnected Ranges: Separable and Low-Density Domains

We prove a Bishop--Phelps--Bollob\'as theorem for operators into spaces of continuous scalar-valued functions on extremally disconnected compact Hausdorff spaces over both the real and complex scalar fields. The main result applies whenever the density character of the domain is strictly smaller than the Baire number of the underlying compact space. The proof also yields an explicit quadratic Bishop--Phelps--Bollob\'as modulus. In particular, every separable Banach space paired with such a function space has the Bishop--Phelps--Bollob\'as property for operators.

math.FA↗

arXiv2026

Classifying Slice-Regular Polynomials via Group Actions on the Twistor Space

We study the equivalence classes of slice-regular functions $f:\Omega\to\mathbb{H}$ on a symmetric slice domain $\Omega$, and of their subclass made of polynomial slice-regular functions, with respect to the natural action of $\mathrm{PGL}(2,\mathbb{H})$ and its subgroups, by employing the twistor construction. In particular, we characterize slice--regular functions whose twistor lift is planar and belongs to a given orbit, and we find normal classes of slice-regular polynomials with respect to the action of a parabolic subgroup of $\mathrm{GL}(2,\mathbb{H})$.

math.DG↗

arXiv2026

SDDF: Specificity-Driven Dynamic Focusing for Open-Vocabulary Camouflaged Object Detection

Open-vocabulary object detection (OVOD) aims to detect known and unknown objects in the open world by leveraging text prompts. Benefiting from the emergence of large-scale vision--language pre-trained models, OVOD has demonstrated strong zero-shot generalization capabilities. However, when dealing with camouflaged objects, the detector often fails to distinguish and localize objects because the visual features of the objects and the background are highly similar. To bridge this gap, we construct a benchmark named OVCOD-D by augmenting carefully selected camouflaged object images with fine-grained textual descriptions. Due to the limited scale of available camouflaged object datasets, we adopt detectors pre-trained on large-scale object detection datasets as our baseline methods, as they possess stronger zero-shot generalization ability. In the specificity-aware sub-descriptions generated by multimodal large models, there still exist confusing and overly decorative modifiers. To mitigate such interference, we design a sub-description principal component contrastive fusion strategy that reduces noisy textual components. Furthermore, to address the challenge that the visual features of camouflaged objects are highly similar to those of their surrounding environment, we propose a specificity-guided regional weak alignment and dynamic focusing method, which aims to strengthen the detector's ability to discriminate camouflaged objects from background. Under the open-set evaluation setting, the proposed method achieves an AP of 56.4 on the OVCOD-D benchmark.

cs.CV↗

arXiv2026

The maximal mean equicontinuous factor via regional mean sensitivity

For actions of amenable groups, mean equicontinuity-a natural relaxation of equicontinuity obtained by averaging metrics along orbits-is well known to yield a maximal mean equicontinuous factor. In 2021, Li and Yu introduced the notion of weak sensitivity in the mean for actions of $\mathbb{Z}$ to gain a deeper understanding of this phenomenon, building on earlier work by Qiu and Zhao. We demonstrate that this relation is insufficient for actions of non-Abelian groups. To overcome this limitation, we introduce the regional mean sensitive relation, which more precisely captures the dynamical behaviour underlying the maximal mean equicontinuous factor. We discuss its fundamental properties and highlight its advantages in the non-Abelian setting. In particular, we show that mean equicontinuity is equivalent to the nonexistence of non-diagonal regional mean sensitive pairs. For this, we work in the context of actions of $\sigma$-compact and locally compact amenable groups.

math.DS↗

arXiv2026

Rigidity of Generalized Furstenberg Boundaries and Applications to Intermediate Crossed Products

We develop a relative boundary theory for actions of discrete groups on compact spaces and use it to derive rigidity results for reduced crossed products. For a discrete group $\Gamma$ acting on a compact space $X$ and a subgroup $H$, we construct a universal boundary over $X$ which is minimal as a $\Gamma$-system and strongly proximal with respect to $H$. When $H\le_c\Gamma$ is commensurated and the $H$-action on $X$ is minimal, we show that this universal boundary agrees, in a canonical $\Gamma$-equivariant way, with the generalized Furstenberg boundary of $(H,X)$, thereby unifying and extending earlier results on relative boundaries. As an application, we introduce the notion of an $X$-plump subgroup given a $\Gamma$-space $X$, a generalized version of plumpness tailored to crossed products. Under natural dynamical hypotheses, this leads to new examples of irreducible $C^*$-inclusions. Under additional assumptions, we also show that every intermediate $C^*$-algebra is a crossed product.

math.OA↗

arXiv2025

Idempotents in the Ellis semigroup of Floyd-Auslander systems

We study minimal idempotents $J^{\mathrm{min}}(X)$ in the Ellis semigroup $E(X)$ associated with a Floyd-Auslander system $(X,T)$. We show that $(X,T)$ is non-tame if and only if $|J^{\mathrm{min}}(X)| > 2^{\aleph_0}$, which happens exactly when the factor map onto the maximal equicontinuous factor possesses uncountably many non-invertible fibres. This yields an easy-to-check criterion for distinguishing tame from non-tame Floyd-Auslander systems and, more importantly, provides an entire family of regular almost automorphic systems with $|J^{\mathrm{min}}(X)| > 2^{\aleph_0}$. Notably, all previously known regular almost automorphic non-tame systems exhibited only a small (i.e. $\leq 2^{\aleph_0}$) set of minimal idempotents. We obtain our result by leveraging an alternative characterisation of (non)-tameness through, what we call, choice domains.

math.DS↗

arXiv2025

A note on multivariate diam mean equicontinuity and frequent stability

Let $(X,G)$ be a topological dynamical system, given by the action of a is a countable discrete infinite group on a compact metric space $X$. We prove that if $(X,G)$ is minimal, then it is either diam-mean $m$-equicontinuious or diam-mean $m$-sensitive. Similarly, $(X,G)$ is either frequently $m$-stable or strongly $m$-spreading. Further, when $G$ is abelian (or, more generally, virtually nilpotent), then the following statements are equivalent: $\bullet$ $(X,G)$ is a regular $m$-to-one extension of its maximal equicontinuous factor; $\bullet$ $(X,G)$ is diam-mean $(m+1)$-equicontinuious, and not diam mean $m$-equicontinuious; $\bullet$ $(X,G)$ is not diam-mean $(m+1)$-sensitive, but diam mean $m$-sensitive; $\bullet$ $(X,G)$ has an essential weakly mean sensitive $m$-tuple but no essential weakly mean sensitive $(m+1)$-tuple. This provides a {\em \enquote*{local}} characterisation of $m$-regularity and mean $m$-sensitivity vial weakly mean sensitive tuples. The same result holds when $G$ is amenable and $(X,G)$ satisfies the local Bronstein condition.

math.DS↗

arXiv2025

The Interplay between Additive and Multiplicative Central Sets Theorems

The concept of Central sets, introduced by Furstenberg through the framework of topological dynamics, has played a pivotal role in combinatorial number theory. Furstenberg's Central Sets Theorem highlighted their rich combinatorial structure. Later, De, Hindman, and Strauss strengthen this theorem using the algebraic framework of the Stone--\v{C}ech compactification. In this article, we establish a unified version of the Central Sets Theorem that simultaneously captures both additive and multiplicative structures.

math.CO↗

arXiv2025

MEBench: Benchmarking Large Language Models for Cross-Document Multi-Entity Question Answering

Multi-entity question answering (MEQA) represents significant challenges for large language models (LLM) and retrieval-augmented generation (RAG) systems, which frequently struggle to consolidate scattered information across diverse documents. While existing methods excel at single-document comprehension, they often struggle with cross-document aggregation, particularly when resolving entity-dense questions like "What is the distribution of ACM Fellows among various fields of study?", which require integrating entity-centric insights from heterogeneous sources (e.g., Wikipedia pages). To address this gap, we introduce MEBench, a novel multi-document, multi-entity benchmark designed to systematically evaluate LLMs' capacity to retrieve, consolidate, and reason over fragmented information. Our benchmark comprises 4,780 questions which are systematically categorized into three primary categories, further divided into eight distinct types, ensuring broad coverage of real-world multi-entity reasoning scenarios. Our experiments on state-of-the-art LLMs (e.g., GPT-4, Llama-3) and RAG pipelines reveal critical limitations: even advanced models achieve only 59% accuracy on MEBench. Our benchmark emphasizes the importance of completeness and factual precision of information extraction in MEQA tasks, using Entity-Attributed F1 (EA-F1) metric for granular evaluation of entity-level correctness and attribution validity. MEBench not only highlights systemic weaknesses in current LLM frameworks but also provides a foundation for advancing robust, entity-aware QA architectures.

cs.CL↗

arXiv2025

Independence and mean sensitivity in minimal systems under group actions

In this paper, we mainly study the relation between regularity, independence and mean sensitivity for minimal systems. In the first part, we show that if a minimal system is incontractible, or local Bronstein with an invariant Borel probability measure, then the regularity is strictly bounded by the infinite independence. In particular, the following two types of minimal systems are applicable to our result: (1) The acting group of the minimal system is a virtually nilpotent group. (2) The minimal system is a proximal extension of its maximal equicontinuous factor and admits an invariant Borel probability measure. Items (1) and (2) correspond to Conjectures 1 and 2 from Huang, Lian, Shao, and Ye (J. Funct. Anal., 2021); item (1) verifies Conjecture 1 in the virtually nilpotent case, and item (2) gives an affirmative answer to Conjecture 2. In the second part, for a minimal system acting by an amenable group, under the local Bronstein condition, we establish parallel results regarding weak mean sensitivity and establish that every mean-sensitive tuple is an IT-tuple.

math.DS↗

arXiv2025

Independence, sequence entropy and mean sensitivity for invariant measures

We investigate the connections between independence, sequence entropy, and mean sensitivity for a measure preserving system under the action of a countable infinite discrete group. We establish that every sequence entropy tuple for an invariant measure is an IT tuple. Furthermore, if the acting group is amenable, we show that for an ergodic measure, the sequence entropy tuples, the mean sensitive tuples along some tempered F{\o}lner sequence, and the sensitive in the mean tuples along some tempered F{\o}lner sequence coincide.

math.DS↗