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Yubin He

Publications and source records attributed to Yubin He.

17 recordsLinked to original sources
arXiv2026

Hausdorff dimension of $\tau$-approximable points on self-similar sets in $\mathbb R^d$

Let $d\geq 1$. Let $K\subset\mathbb{R}^d$ be a non-singleton self-similar set generated by a finite strongly irreducible iterated function system satisfying the open set condition, and let $\delta=\dim_{\mathrm H} K$. For $\tau>1/d$, set \[ W_d(\tau) = \left\{ \mathbf{x}\in\mathbb{R}^d: |q\mathbf{x}-\mathbf{p}| 0$ such that, for every $1/d<\tau<1/d+\varepsilon_K$, \[ \mathcal{H}^{s(\tau)}(K\cap W_d(\tau))=\infty, \qquad\text{with } s(\tau):=\delta+\frac{d+1}{1+\tau}-d, \] and consequently \[ \dim_{\mathrm H}(K\cap W_d(\tau)) = \delta+\frac{d+1}{1+\tau}-d. \] In dimension one, specializing to the middle-third Cantor set, this establishes the Bugeaud--Durand conjectural formula for $\tau>1$ sufficiently close to $1$.

math.NT↗
arXiv2026

DLIOS: An LLM-Augmented Real-Time Multi-Modal Interactive Enhancement Overlay System for Douyin Live Streaming

We present DLIOS, a Large Language Model (LLM)-augmented real-time multi-modal interactive enhancement overlay system for Douyin (TikTok) live streaming. DLIOS employs a three-layer transparent window architecture for independent rendering of danmaku (scrolling text), gift and like particle effects, and VIP entrance animations, built around an event-driven WebView2 capture pipeline and a thread-safe event bus. On top of this foundation we contribute an LLM broadcast automation framework comprising: (1) a per-song four-segment prompt scheduling system (T1 opening/transition, T2 empathy, T3 era story/production notes, T4 closing) that generates emotionally coherent radio-style commentary from lyric metadata; (2) a JSON-serializable RadioPersonaConfig schema supporting hot-swap multi-persona broadcasting; (3) a real-time danmaku quick-reaction engine with keyword routing to static urgent speech or LLM-generated empathetic responses; and (4) the Suwan Li AI singer-songwriter persona case study -- over 100 AI-generated songs produced with Suno. A 36-hour stress test demonstrates: zero danmaku overlap, zero deadlock crashes, gift effect P95 latency <= 180 ms, LLM-to-TTS segment P95 latency <= 2.1 s, and TTS integrated loudness gain of 9.5 LUFS. live streaming; danmaku; large language model; prompt engineering; virtual persona; WebView2; WINMM; TTS; Suno; loudness normalization; real-time scheduling

eess.IV↗
arXiv2026

Jarn\'ik-type theorem for self-similar sets

Let $K\subset\mathbb R^d$ be a compact subset equipped with a $\delta$-Ahlfors regular measure $\mu$. For any $\tau>1/d$ and any ``inhomogeneous'' vector $\boldsymbol{\theta}\in\mathbb R^d$, let $W_d(\psi_\tau,\boldsymbol{\theta})$ denote the set of $(\psi_\tau,\boldsymbol{\theta})$-well approximable numbers, where $\psi_\tau(q)=q^{-\tau}$. Assuming a local estimate for the $\mu$-measure of the intersections of $K$ with the neighborhoods of ``rational'' vectors $(\mathbf p+\boldsymbol{\theta})/q$, we establish a sharp upper bound for the Hausdorff dimension of $K\cap W_d(\psi_\tau,\boldsymbol{\theta})$, together with some nontrivial lower bounds when $\tau$ is below a certain threshold. One of the lower bounds becomes sharp in the one-dimensional homogeneous case ($d=1$, $\theta=0$) for a class of sufficiently thick self-similar sets $K$, and moreover $K\cap W_1(\psi_\tau,0)$ has full $(\delta+\frac{2}{1+\tau}-1)$-Hausdorff measure. These results have several applications: (1) the set of homogeneous very well approximable numbers has full Hausdorff dimension within strongly irreducible self-similar sets in $\mathbb R^d$, extending a recent result of Chen [arXiv:2510.17096]; (2) the set of inhomogeneous very well approximable numbers has full Hausdorff dimension within sufficiently thick missing digits sets in $\mathbb R$, affirmatively answering a question posed by Yu [arXiv:2101.05910]. Our applications build on the seminal works of Yu [arXiv:2101.05910] and B\'enard, He and Zhang [arXiv:2508.09076]. We also provide some non-trivial missing digits set $K\subset[0,1]^d$ whose intersection with $W_d(\psi_\tau,0)$ has full $(\delta+\frac{1+d}{1+\tau}-d)$-Hausdorff measure.

math.NT↗
arXiv2025

Evaluating Hydro-Science and Engineering Knowledge of Large Language Models

Hydro-Science and Engineering (Hydro-SE) is a critical and irreplaceable domain that secures human water supply, generates clean hydropower energy, and mitigates flood and drought disasters. Featuring multiple engineering objectives, Hydro-SE is an inherently interdisciplinary domain that integrates scientific knowledge with engineering expertise. This integration necessitates extensive expert collaboration in decision-making, which poses difficulties for intelligence. With the rapid advancement of large language models (LLMs), their potential application in the Hydro-SE domain is being increasingly explored. However, the knowledge and application abilities of LLMs in Hydro-SE have not been sufficiently evaluated. To address this issue, we propose the Hydro-SE LLM evaluation benchmark (Hydro-SE Bench), which contains 4,000 multiple-choice questions. Hydro-SE Bench covers nine subfields and enables evaluation of LLMs in aspects of basic conceptual knowledge, engineering application ability, and reasoning and calculation ability. The evaluation results on Hydro-SE Bench show that the accuracy values vary among 0.74 to 0.80 for commercial LLMs, and among 0.41 to 0.68 for small-parameter LLMs. While LLMs perform well in subfields closely related to natural and physical sciences, they struggle with domain-specific knowledge such as industry standards and hydraulic structures. Model scaling mainly improves reasoning and calculation abilities, but there is still great potential for LLMs to better handle problems in practical engineering application. This study highlights the strengths and weaknesses of LLMs for Hydro-SE tasks, providing model developers with clear training targets and Hydro-SE researchers with practical guidance for applying LLMs.

cs.CL↗
arXiv2025

Shrinking Targets versus Recurrence: a brief survey

Let $(X,d)$ be a compact metric space and $(X,\mathcal{A},\mu,T)$ a measure preserving dynamical system. Furthermore, given a real, positive function $\psi$, let $W(T, \psi)$ and $ R(T,\psi) $ respectively denote the shrinking target set and the recurrent set associated with the dynamical system. Under certain mixing properties it is known that if the natural measure sum diverges then the recurrent and shrinking target sets are of full $\mu$-measure. The purpose of this survey is to provide a brief overview of such results, to discuss the potential quantitative strengthening of the full measure statements and to bring to the forefront key differences in the theory.

math.DS↗
arXiv2025

A dimensional mass transference principle from balls to open sets and applications to dynamical Diophantine approximation

The mass transference principle of Beresnevich and Velani is a powerful mechanism for determining the Hausdorff dimension/measure of $\limsup$ sets that arise naturally in Diophantine approximation. However, in the setting of dynamical Diophantine approximation, this principle often fails to apply effectively, as the radii of the balls defining the dynamical $\limsup$ sets generally depend on the orbit of the point $x$ itself. In this paper, we develop a dimensional mass transference principle that enables us to recover and extend classical results on shrinking target problems, particularly for the $\beta$-transformation and the Gauss map. Moreover, our result shows that the corresponding $\limsup$ sets have large intersection properties. A potentially interesting feature of our method is that, in many cases, shrinking target problems are closely related to finding an appropriate Gibbs measure, which may reveal new aspects of the link between thermodynamic formalism and dynamical Diophantine approximation.

math.NT↗
arXiv2025

CRIA: A Cross-View Interaction and Instance-Adapted Pre-training Framework for Generalizable EEG Representations

The difficulty of extracting deep features from EEG data and effectively integrating information from multiple views presents significant challenges for developing a generalizable pretraining framework for EEG representation learning. However, most existing pre-training methods rely solely on the contextual semantics of a single view, failing to capture the complex and synergistic interactions among different perspectives, limiting the expressiveness and generalization of learned representations. To address these issues, this paper proposes CRIA, an adaptive framework that utilizes variable-length and variable-channel coding to achieve a unified representation of EEG data across different datasets. In this work, we define cross-view information as the integrated representation that emerges from the interaction among temporal, spectral, and spatial views of EEG signals. The model employs a cross-attention mechanism to fuse temporal, spectral, and spatial features effectively, and combines an attention matrix masking strategy based on the information bottleneck principle with a novel viewpoint masking pre-training scheme. Experimental results on the Temple University EEG corpus and the CHB-MIT dataset show that CRIA outperforms existing methods with the same pre-training conditions, achieving a balanced accuracy of 57.02% for multi-class event classification and 80.03% for anomaly detection, highlighting its strong generalization ability.

cs.LG↗
arXiv2025

Hausdorff measure and Fourier dimensions of limsup sets arising in weighted and multiplicative Diophantine approximation

The classical Khintchine--Jarn\'ik Theorem provides elegant criteria for determining the Lebesgue measure and Hausdorff measure of sets of points approximated by rational points, which has inspired much modern research in metric Diophantine approximation. This paper concerns the Lebesgue measure, Hausdorff measure and Fourier dimension of sets arising in weighted and multiplicative Diophantine approximation. We provide zero-full laws for determining the Lebesgue measure and Hausdorff measure of the sets under consideration. In particular, the criterion for the weighted setup refines a dimensional result given by Li, Liao, Velani, Wang, and Zorin [arXiv: 2410.18578 (2024)], while the criteria for the multiplicative setup answer a question raised by Hussain and Simmons [J. Number Theory (2018)] and extend beyond it. A crucial observation is that, even in higher dimensions, both setups are more appropriately understood as consequences of the `balls-to-rectangles' mass transference principle. We also determine the exact Fourier dimensions of these sets. The result we obtain indicates that, in line with the existence results, these sets are generally non-Salem sets, except in the one-dimensional case. This phenomenon can be partly explained by another result of this paper, which states that the Fourier dimension of the product of two sets equals the minimum of their respective Fourier dimensions.

math.NT↗
arXiv2024

Dichotomy laws for the Hausdorff measure of shrinking target sets in $\beta$-dynamical systems

In this paper, we investigate the Hausdorff measure of shrinking target sets in $\beta$-dynamical systems. These sets are dynamically defined in analogy to the classical theory of weighted and multiplicative approximation. While the Lebesgue measure and Hausdorff dimension theories for these sets are well-understood, the Hausdorff measure theory in even one-dimensional settings remains unknown. We show that the Hausdorff measure of these sets is either zero or full depending upon the convergence or divergence of a certain series, thus providing a rather complete measure theoretic description of these sets.

math.DS↗
arXiv2024

Quantitative recurrence properties and strong dynamical Borel-Cantelli lemma for dynamical systems with exponential decay of correlations

Let $ ([0,1]^d,T,\mu) $ be a measure-preserving dynamical system so that the correlations decay exponentially for H\"older continuous functions. Suppose that $ \mu $ is absolutely continuous with a density function $ h\in L^q(\mathcal L^d) $ for some $ q>1 $, where $ \mathcal L^d $ is the $ d $-dimensional Lebesgue measure. Under mild conditions on the underlying dynamical system, we obtain a strong dynamical Borel-Cantelli lemma for recurrence: For any sequence $ \{R_n\} $ of hyperrectangles with sides parallel to the axes and centered at the origin, \[\sum_{n=1}^{\infty}\mathcal L^d(R_n)=\infty\quad\Longrightarrow\quad\lim_{n\to\infty}\frac{\sum_{k=1}^{n}\chi_{R_k+\mathbf{x}}(T^k\mathbf{x})}{\sum_{k=1}^{n}\mathcal L^d(R_k)}=h(\mathbf{x})\quad\text{for $ \mu $-a.e.$\textbf{x}$},\] where $ \textbf{x}\in[0,1]^d $ and $ R_k+\textbf{x} $ is the translation of $ R_k $. The result applies to Gauss map, $\beta$-transformation and expanding toral endomorphisms.

math.DS↗
arXiv2024

A unified approach to mass transference principle and large intersection property

The mass transference principle, discovered by Beresnevich and Velani [Ann Math (2), 2006], is a landmark result in Diophantine approximation that allows us to obtain the Hausdorff measure theory of $\limsup$ set. Another important tool is the notion of large intersection property, introduced and systematically studied by Falconer [J. Lond. Math. Soc. (2), 1994]. The former mainly focuses on passing between full (Lebesgue) measure and full Hausdorff measure statements, while the latter transfers full Hausdorff content statement to Hausdorff dimension. From this perspective, the proofs of the two results are quite similar but often treated in different ways. In this paper, we establish a general mass transference principle from the viewpoint of Hausdorff content, aiming to provide a unified proof for the aforementioned results. More precisely, this principle allows us to transfer the Hausdorff content bounds of a sequence of open sets $E_n$ to the full Hausdorff measure statement and large intersection property for $\limsup E_n$. One of the advantages of our approach is that the verification of the Hausdorff content bound does not require the construction of Cantor-like subset, resulting in a much simpler proof. As an application, we provide simpler proofs for several mass transference principles.

math.NT↗
arXiv2023

Shrinking parallelepiped targets in beta-dynamical systems

For $ \beta>1 $ let $ T_\beta $ be the $\beta$-transformation on $ [0,1) $. Let $ \beta_1,\dots,\beta_d>1 $ and let $ \mathcal P=\{P_n\}_{n\ge 1} $ be a sequence of parallelepipeds in $ [0,1)^d $. Define \[W(\mathcal P)=\{\textbf{x}\in[0,1)^d:(T_{\beta_1}\times\cdots \times T_{\beta_2})^n(\textbf{x})\in P_n\text{ infinitely often}\}.\] When each $ P_n $ is a hyperrectangle with sides parallel to the axes, the 'rectangle to rectangle' mass transference principle by Wang and Wu [Math. Ann. 381 (2021)] is usually employed to derive the lower bound for $\mathrm{dim_H} W(\mathcal P)$, where $\mathrm{dim_H}$ denotes the Hausdorff dimension. However, in the case where $ P_n $ is still a hyperrectangle but with rotation, this principle, while still applicable, often fails to yield the desired lower bound. In this paper, we determine the optimal cover of parallelepipeds, thereby obtaining $\mathrm{dim_H} W(\mathcal P)$. We also provide several examples to illustrate how the rotations of hyperrectangles affect $\mathrm{dim_H} W(\mathcal P)$.

math.DS↗
arXiv2023

Quantitative recurrence properties for piecewise expanding maps on $ [0,1]^d $

Let $ T\colon[0,1]^d\to [0,1]^d $ be a piecewise expanding map with an absolutely continuous invariant measure $ μ$. Let $ \{H_n\} $ be a sequence of hyperrectangles or hyperboloids centered at the origin. Denote by $ \mathcal R(\{H_n\}) $ the set of points $ \mathbf x $ such that $ T^n\mathbf x\in \mathbf x+H_n $ for infinitely many $ n\in\mathbb N $, where $ \mathbf x+H_n $ is the translation of $ H_n $. We prove that if $ μ$ is exponential mixing and the density of $ μ$ is sufficiently regular, then the $μ$-measure of $ \mathcal R(\{H_n\}) $ is zero or full according to the sum of the volumes of $ H_n $ converges or not. In the case that $ T $ is a matrix transformation, our results extend a previous work of Kirsebom, Kunde, and Persson [to appear in Ann. Sc. Norm. Super. Pisa Cl. Sci., 2023] in two aspects: by allowing the matrix to be non-integer and by allowing the `target' sets $ H_n $ to be hyperrectangles or hyperboloids. We also obtain a dimension result when $ T $ is a diagonal matrix transformation.

math.DS↗
arXiv2022

Uniform approximation problems of expanding Markov maps

Let $ T:[0,1]\to[0,1] $ be an expanding Markov map with a finite partition. Let $ μ_ϕ$ be the invariant Gibbs measure associated with a Hölder continuous potential $ ϕ$. In this paper, we investigate the size of the uniform approximation set \[\mathcal U^κ(x):=\{y\in[0,1]:\forall N\gg1,~\exists n\le N, \text{ such that }|T^nx-y| 0 $ and $ x\in[0,1] $. The critical value of $ κ$ such that $ \textrm{dim}_{\textrm H}\mathcal U^κ(x)=1 $ for $ μ_ϕ$-a.e.$ \, x $ is proven to be $ 1/α_{\max} $, where $ α_{\max}=-\int ϕ\,dμ_{\max}/\int\log|T'|\,dμ_{\max} $ and $ μ_{\max} $ is the Gibbs measure associated with the potential $ -\log|T'| $. Moreover, when $ κ>1/α_{\max} $, we show that for $ μ_ϕ$-a.e.$ \, x $, the Hausdorff dimension of $ \mathcal U^κ(x) $ agrees with the multifractal spectrum of $ μ_ϕ$.

math.DS↗
arXiv2021

Sets of Exact Approximation Order by Complex rational numbers

For a nonincreasing function $ψ$, let $\textrm{Exact}(ψ)$ be the set of complex numbers that are approximable by complex rational numbers to order $ψ$ but to no better order. In this paper, we obtain the Hausdorff dimension and packing dimension of $\textrm{Exact}(ψ)$ when $ψ(x)=o(x^{-2})$. We also prove that the lower bound of the Hausdorff dimension is greater than $2-τ/(1-2τ)$ when $τ=\limsup_{x\to\infty}ψ(x)x^2$ small enough.

math.NT↗
arXiv2021

The difference between the Hurwitz continued fraction expansions of a complex number and its rational approximations

For regular continued fraction, if a real number $x$ and its rational approximation $p/q$ satisfying $|x-p/q|<1/q^2$, then, after deleting the last integer of the partial quotients of $p/q$, the sequence of the remaining partial quotients is a prefix of that of $x$. In this paper, we show that the situation is completely different if we consider the Hurwitz continued fraction expansions of a complex number and its rational approximations. More specifically, we consider the set $E(ψ)$ of complex numbers which are well approximated with the given bound $ψ$ and have quite different Hurwitz continued fraction expansions from that of their rational approximations. The Hausdorff and packing dimensions of such set are determined. It turns out that its packing dimension is always full for any given approximation bound $ψ$ and its Hausdorff dimension is equal to that of the $ψ$-approximable set $W(ψ)$ of complex numbers. As a consequence, we also obtain an analogue of the classical Jarník Theorem in real case.

math.NT↗