arXiv subjects

Kan Jiang

Publications and source records attributed to Kan Jiang.

At least 19 recordsLinked to original sources

arXiv2026

TennisExpert: Towards Expert-Level Analytical Sports Video Understanding

Tennis is one of the most widely followed sports, generating extensive broadcast footage with strong potential for professional analysis, automated coaching, and real-time commentary. However, automatic tennis understanding remains underexplored due to two key challenges: (1) the lack of large-scale benchmarks with fine-grained annotations and expert-level commentary, and (2) the difficulty of building accurate yet efficient multimodal systems suitable for real-time deployment. To address these challenges, we introduce TennisVL, a large-scale tennis benchmark comprising over 200 professional matches (471.9 hours) and 40,000+ rally-level clips. Unlike existing commentary datasets that focus on descriptive play-by-play narration, TennisVL emphasizes expert analytical commentary capturing tactical reasoning, player decisions, and match momentum. Furthermore, we propose TennisExpert, a multimodal tennis understanding framework that integrates a video semantic parser with a memory-augmented model built on Qwen3-VL-8B. The parser extracts key match elements (e.g., scores, shot sequences, ball bounces, and player locations), while hierarchical memory modules capture both short- and long-term temporal context. Experiments show that TennisExpert consistently outperforms strong proprietary baselines, including GPT-5, Gemini, and Claude, and demonstrates improved ability to capture tactical context and match dynamics. Our dataset and code are publicly available at https://github.com/LZYAndy/TennisExpert.

cs.CV↗

arXiv2025

Few-Shot Precise Event Spotting via Unified Multi-Entity Graph and Distillation

Precise event spotting (PES) aims to recognize fine-grained events at exact moments and has become a key component of sports analytics. This task is particularly challenging due to rapid succession, motion blur, and subtle visual differences. Consequently, most existing methods rely on domain-specific, end-to-end training with large labeled datasets and often struggle in few-shot conditions due to their dependence on pixel- or pose-based inputs alone. However, obtaining large labeled datasets is practically hard. We propose a Unified Multi-Entity Graph Network (UMEG-Net) for few-shot PES. UMEG-Net integrates human skeletons and sport-specific object keypoints into a unified graph and features an efficient spatio-temporal extraction module based on advanced GCN and multi-scale temporal shift. To further enhance performance, we employ multimodal distillation to transfer knowledge from keypoint-based graphs to visual representations. Our approach achieves robust performance with limited labeled data and significantly outperforms baseline models in few-shot settings, providing a scalable and effective solution for few-shot PES. Code is publicly available at https://github.com/LZYAndy/UMEG-Net.

cs.CV↗

arXiv2025

On the intersection of Cantor set with the unit circle and some sequences

For $\lambda\in(0,1/2)$ let $K_\lambda$ be the self-similar set in $\mathbb{R}$ generated by the iterated function system $\{f_0(x)=\lambda x, f_1(x)=\lambda x+1-\lambda \}$. In this paper, we investigate the intersection of the unit circle $\mathbb{S} \subset \mathbb{R}^2$ with the Cartesian product $K_{\lambda} \times K_{\lambda}$. We prove that for $\lambda \in(0, 2 - \sqrt{3}]$, the intersection is trivial, i.e., \[ \mathbb{S} \cap (K_{\lambda} \times K_{\lambda}) = \{(0,1), (1,0)\}. \] If $\lambda\in [0.330384,1/2)$, then the intersection $\mathbb{S} \cap (K_{\lambda} \times K_{\lambda})$ is non-trivial. In particular, if $\lambda\in [0.407493 , 1/2)$ the intersection $\mathbb{S} \cap (K_{\lambda} \times K_{\lambda})$ is of cardinality continuum. Furthermore, the bound $2 - \sqrt{3}$ is sharp: there exists a sequence $\{\lambda_n\}_{n \in \mathbb{N}}$ with $\lambda_n \searrow 2 - \sqrt{3}$ such that $\mathbb{S} \cap (K_{\lambda_n} \times K_{\lambda_n})$ is non-trivial for all $n\in\mathbb{N}$. This result provides a negative answer to a problem posed by Yu (2023). Our methods extend beyond the unit circle and remain effective for many nonlinear curves. By employing tools from number theory, including the quadratic reciprocity law, we analyze the intersection of Cantor sets with some sequences. A dichotomy is established in terms of the Legendre symbol associated with the digit set, revealing a fundamental arithmetic constraint governing such intersections.

math.CA↗

arXiv2025

F$^3$Set: Towards Analyzing Fast, Frequent, and Fine-grained Events from Videos

Analyzing Fast, Frequent, and Fine-grained (F$^3$) events presents a significant challenge in video analytics and multi-modal LLMs. Current methods struggle to identify events that satisfy all the F$^3$ criteria with high accuracy due to challenges such as motion blur and subtle visual discrepancies. To advance research in video understanding, we introduce F$^3$Set, a benchmark that consists of video datasets for precise F$^3$ event detection. Datasets in F$^3$Set are characterized by their extensive scale and comprehensive detail, usually encompassing over 1,000 event types with precise timestamps and supporting multi-level granularity. Currently, F$^3$Set contains several sports datasets, and this framework may be extended to other applications as well. We evaluated popular temporal action understanding methods on F$^3$Set, revealing substantial challenges for existing techniques. Additionally, we propose a new method, F$^3$ED, for F$^3$ event detections, achieving superior performance. The dataset, model, and benchmark code are available at https://github.com/F3Set/F3Set.

cs.CV↗

arXiv2022

Thickness theorems with partial derivatives

In this paper, we prove some new thickness theorems with partial derivatives. We give some applications. First, we give a simple criterion that can judge whether two scaled Cantor sets have non-empty intersection. Second, we prove under some checkable conditions that the continuous image of arbitrary self-similar sets with positive similarity ratios is a closed interval, a finite union of closed intervals or containing interior. Third, we prove an analogous Erdős-Straus conjecture on the middle-third Cantor set. Finally, we consider the solutions to the Diophantine equations on fractal sets. More specifically, for various Diophantine equations, we cannot find a solution on certain self-similar sets, whilst for the Fermat's equation, which is associated with the famous Fermat's last theorem, we can find infinitely many solutions on many self-similar sets.

math.DS↗

arXiv2022

Rational Points in Translations of The Cantor Set

Given two coprime integers $p\ge 2$ and $q \ge 3$, let $D_p\subset[0,1)$ consist of all rational numbers which have a finite $p$-ary expansion, and let $$ K(q, \mathcal{A})=\bigg\{ \sum_{i=1}^\infty \frac{d_i}{q^i}: d_i\in \mathcal{A}~ \forall i\in\mathbb{N} \bigg\}, $$ where $\mathcal{A} \subset \{0,1,\ldots, q-1\}$ with cardinality $1<\#\mathcal{A}< q$. In 2021 Schleischitz showed that $\#(D_p\cap K(q,\mathcal{A}))<+\infty$. In this paper we show that for any $r\in\mathbb{Q}$ and for any $\alpha\in\mathbb{R}$, $$ \#\big((r D_p+\alpha)\cap K(q,\mathcal{A})\big)<+\infty. $$

math.NT↗

arXiv2022

Irrational self-similar sets

Let $K\subset\mathbb{R}$ be a self-similar set defined on $\mathbb{R}$. It is easy to prove that if the Lebesgue measure of $K$ is zero, then for Lebesgue almost every $t$, $$K+t=\{x+t:x\in K\}$$ only consists of irrational or transcendental numbers. In this note, we shall consider some classes of self-similar sets, and explicitly construct such $t$'s. Our main idea is from the $q$-expansions.

math.NT↗

arXiv2022

On some fractal sets only containing irrationals

In this paper, we prove that many fractal sets generated by the associated dynamical systems only contain irrationals. As an application, we explicitly construct some overlapping self-similar sets which only consist of irrationals.

math.NT↗

arXiv2021

On a class of self-similar sets which contain finitely many common points

For $\lambda\in(0,1/2]$ let $K_\lambda \subset\mathbb{R}$ be a self-similar set generated by the iterated function system $\{\lambda x, \lambda x+1-\lambda\}$. Given $x\in(0,1/2)$, let $\Lambda(x)$ be the set of $\lambda\in(0,1/2]$ such that $x\in K_\lambda$. In this paper we show that $\Lambda(x)$ is a topological Cantor set having zero Lebesgue measure and full Hausdorff dimension. Furthermore, we show that for any $y_1,\ldots, y_p\in(0,1/2)$ there exists a full Hausdorff dimensional set of $\lambda\in(0,1/2]$ such that $y_1,\ldots, y_p \in K_\lambda$.

math.DS↗

arXiv2021

How likely can a point be in different Cantor sets

Let $m\in\mathbb N_{\ge 2}$, and let $\mathcal K=\{K_λ: λ\in(0, 1/m]\}$ be a class of Cantor sets, where $K_λ=\{\sum_{i=1}^\infty d_iλ^i: d_i\in\{0,1,\ldots, m-1\}, i\ge 1\}$. We investigate in this paper the likelyhood of a fixed point in the Cantor sets of $\mathcal K$. More precisely, for a fixed point $x\in(0,1)$ we consider the parameter set $Λ(x)=\{λ\in(0,1/m]: x\in K_λ\}$, and show that $Λ(x)$ is a topological Cantor set having zero Lebesgue measure and full Hausdorff dimension. Furthermore, by constructing a sequence of Cantor subsets with large thickness in $Λ(x)$ we prove that the intersection $Λ(x)\capΛ(y)$ also has full Hausdorff dimension for any $x, y\in(0,1)$.

math.DS↗

arXiv2020

A nonlinear version of the Newhouse thickness theorem

Let $C_1$ and $C_2$ be two Cantor sets with convex hull $[0,1]$. Newhouse proved if $τ(C_1)\cdot τ(C_2)\geq 1$, then the arithmetic sum $C_1+C_2$ is an interval, where $τ(C_i), 1\leq i\leq 2$ denotes the thickness of $C_i$. In this paper, we generalize this thickness theorem as follows. Let $K_i\subset \mathbb{R}, i=1,\cdots, d$, be some Cantor sets (perfect and nowhere dense) with convex hull $[0,1]$. Suppose $f(x_1,\cdots, x_{d-1},z)\in \mathcal{C}^1$ is a continuous function defined on $\mathbb{R}^d$. Denote the continuous image of $f$ by $$f(K_1,\cdots, K_d)=\{f(x_1, \cdots x_{d-1},z):x_i\in K_i,z\in K_d, 1\leq i\leq d-1\}.$$ If for any $(x_1, \cdots, x_{d-1},z)\in [0,1]^d$, we have $$(τ(K_i))^{-1}\leq \left|\dfrac{\partial_{x_i} f}{\partial_z f}\right|\leq τ(K_d),1\leq i\leq d-1$$ then $f(K_1,\cdots, K_d)$ is a closed interval. We give two applications. Firstly, we partially answer some questions posed by Takahashi. Secondly, we obtain various nonlinear identities, associated with the continued fractions with restricted partial quotients, which can represent real numbers.

math.DS↗

arXiv2020

On continuous images of self-similar sets

Let $(\mathcal{M}, c_k, n_k,κ)$ be a class of homogeneous Moran sets. Suppose $f(x,y)\in C^3$ is a function defined on $\mathbb{R}^2$. Given $E_1, E_2\in(\mathcal{M}, c_k, n_k,κ) $, in this paper, we prove, under some checkable conditions on the partial derivatives of $f(x,y)$, that $$f(E_1,E_2)=\{f(x,y):x\in E_1,y\in E_2\}$$ is exactly a closed interval or a union of finitely many closed intervals. Similar results for the homogeneous self-similar sets with arbitrary overlaps can be obtained. Further generalization is available for some inhomogeneous self-similar sets if we utilize the approximation theorem.

math.MG↗

arXiv2020

On the sum of squares of middle-third Cantor set

Let $C$ be the middle-third Cantor set. In this paper, we show that for every $x\in [0,4]$, there exist $x_1, x_2, x_3, x_4 \in C$ such that $$x= x_1^2+x_2^2+x_3^2+x_4^2,$$ which answers a question posed by Athreya, Reznick,and Tyson.

math.DS↗

arXiv2019

Multiplication on uniform $λ$-Cantor sets

Let $C$ be the middle-third Cantor set. Define $C*C=\{x*y:x,y\in C\}$, where $*=+,-,\cdot,÷$ (when $*=÷$, we assume $y\neq0$). Steinhaus \cite{HS} proved in 1917 that \[ C-C=[-1,1], C+C=[0,2]. \] In 2019, Athreya, Reznick and Tyson \cite{Tyson} proved that \[ C÷C=\bigcup_{n=-\infty}^{\infty}\left[ 3^{-n}\dfrac{2}{3},3^{-n}\dfrac {3}{2}\right] . \] In this paper, we give a description of the topological structure and Lebesgue measure of $C\cdot C$. We indeed obtain corresponding results on the uniform $λ$-Cantor sets.

math.DS↗

arXiv2019

Arithmetic on self-similar sets

Let $K_1$ and $K_2$ be two one-dimensional homogeneous self-similar sets. Let $f$ be a continuous function defined on an open set $U\subset \mathbb{R}^{2}$. Denote the continuous image of $f$ by $$ f_{U}(K_1,K_2)=\{f(x,y):(x,y)\in (K_1\times K_2)\cap U\}. $$ In this paper we give an sufficient condition which guarantees that $f_{U}(K_1,K_2)$ contains some interiors. Our result is different from Simon and Taylor's \cite[Proposition 2.9]{ST} as we do not need the condition that the multiplication of the thickness of $K_1$ and $K_2$ is strictly greater than $1$. As a consequence, we give an application to the univoque sets in the setting of $q$-expansions.

math.DS↗

arXiv2019

Multiple codings for self-similar sets with overlaps

In this paper we consider a general class $\mathcal E$ of self-similar sets with complete overlaps. Given a self-similar iterated function system $Φ=(E, \{f_i\}_{i=1}^m)\in\mathcal E$ on the real line, for each point $x\in E$ we can find a sequence $(i_k)=i_1i_2\ldots\in\{1,\ldots,m\}^\mathbb N$, called a coding of $x$, such that $$ x=\lim_{n\to\infty}f_{i_1}\circ f_{i_{2}}\circ\cdots\circ f_{i_n}(0). $$ For $k=1,2,\ldots, \aleph_0$ or $2^{\aleph_0}$ we investigate the subset $\mathcal U_k(Φ)$ which consists of all $x\in E$ having precisely $k$ different codings. Among several equivalent characterizations we show that $\mathcal U_1(Φ)$ is closed if and only if $\mathcal U_{\aleph_0}(Φ)$ is an empty set. Furthermore, we give explicit formulae for the Hausdorff dimension of $\mathcal U_k(Φ)$, and show that the corresponding Hausdorff measure of $\mathcal U_k(Φ)$ is always infinite for any $k\ge 2$. Finally, we explicitly calculate the local dimension of the self-similar measure at each point in $\mathcal U_k(Φ)$ and ${U_{\aleph_0}(Φ)}$.

math.DS↗

arXiv2019

Arithmetic on Moran sets

Let $(\mathcal{M}, c_k,n_k)$ be a class of Moran sets. We assume that the convex hull of any $E\in (\mathcal{M}, c_k,n_k)$ is $[0,1]$. Let $A,B$ be two non-empty sets in $\mathbb{R}$. Suppose that $f$ is a continuous function defined on an open set $U\subset \mathbb{R}^{2}$. Denote the continuous image of $f$ by \begin{equation*} f_{U}(A,B)=\{f(x,y):(x,y)\in (A\times B)\cap U\}. \end{equation*} In this paper, we prove the following result. Let $E_1,E_2\in(\mathcal{M}, c_k, n_k)$. If there exists some $(x_0,y_0)\in (E_1\times E_2)\cap U$ such that $$\sup_{k\geq 1}\left\{1-c_kn_k\right\}<\left\vert \frac{\partial _{y}f|_{(x_{0},y_{0})}}{\partial _{x}f|_{(x_{0},y_{0})}}\right\vert <\inf_{k\geq 1}\left\{\dfrac{c_k}{1-n_kc_k}\right\},$$ then $f_U(E_1, E_2)$ contains an interior.

math.DS↗

arXiv2019

Visibility of Cartesian products of Cantor sets

Let $K_λ$ be the attractor of the following IFS \begin{equation*} \{f_1(x)=λx, f_2(x)=λx+1-λ\}, \;\;0<λ<1/2. \end{equation*} Given $α\geq 0$, we say the line $y=αx$ is visible through $K_λ\times K_λ$ if $$ \{(x, αx): x\in \mathbb R\setminus \{0\}\}\cap ((K_λ\times K_λ))=\emptyset. $$ Let $V=\left \{α\geq 0: y=αx \mbox{ is visible through } K_λ\times K_λ \right \}$. In this paper, we give a completed description of $V$, e.g., its Hausdoff dimension and its topological property. Moreover, we also discuss another type of visible problem which is related to the slicing problem.

math.DS↗