arXiv subjects
Publications and source records attributed to Runlin Zhang.
We study the effective equidistribution of certain infinite homogeneous measures and related counting problems through mixing. In this way, we obtain smooth versions of counting theorems studied by Oh-Shah and later by Kelmer-Kontorovich over a number field. In the appendix, we apply the meromorphic continuation of Hilbert-Asai Eisenstein series to obtain the authentic counting.
Let $k$ be a number field. Let $q(x_1,\cdots,x_n)$ be a non-degenerate integral quadratic form in $n\geq 3$ variables with coefficients in $k$ and $m\in k^\times$. Let $X$ be the affine quadric defined by $q=m$ in $\mathbb{A}^n_k$. Based on results on effective equidistribution of $S$-integral points in symmetric spaces, we establish the following: (i) The arithmetic purity of strong approximation off any single place of $k$ for $X$; (ii) The geometric sieve for $p_0$-integral points on $X$ when $k=\mathbb{Q}$ and $p_0$ is a prime number.
We present MarkMatch, a retrieval system for detecting whether two paper ballot marks were filled by the same hand. Unlike the previous SOTA method BubbleSig, which used binary classification on isolated mark pairs, MarkMatch ranks stylistic similarity between a query mark and a mark in the database using contrastive learning. Our model is trained with a dense batch similarity matrix and a dual loss objective. Each sample is contrasted against many negatives within each batch, enabling the model to learn subtle handwriting difference and improve generalization under handwriting variation and visual noise, while diagonal supervision reinforces high confidence on true matches. The model achieves an F1 score of 0.943, surpassing BubbleSig's best performance. MarkMatch also integrates Segment Anything Model for flexible mark extraction via box- or point-based prompts. The system offers election auditors a practical tool for visual, non-biometric investigation of suspicious ballots.
Hallucinations in vision-language models (VLMs) hinder reliability and real-world applicability, usually stemming from distribution shifts between pretraining data and test samples. Existing solutions, such as retraining or fine-tuning on additional data, demand significant computational resources and labor-intensive data collection, while ensemble-based methods incur additional costs by introducing auxiliary VLMs. To address these challenges, we propose a novel test-time adaptation framework using reinforcement learning to mitigate hallucinations during inference without retraining or any auxiliary VLMs. By updating only the learnable parameters in the layer normalization of the language model (approximately 0.003% of the model parameters), our method reduces distribution shifts between test samples and pretraining samples. A CLIP-based hallucination evaluation model is proposed to provide dual rewards to VLMs. Experimental results demonstrate a 15.4% and 17.3% reduction in hallucination rates on LLaVA and InstructBLIP, respectively. Our approach outperforms state-of-the-art baselines with a 68.3% improvement in hallucination mitigation, demonstrating its effectiveness.
Let U be a homogeneous variety over Q of a linear algebraic group. Choose an integral model and assume the existence of infinitely many integral points. Then one would like to give an asymptotic count of integral points of bounded height with the help of some height function. In many cases, with the help of measure rigidity of unipotent flows, we reduce this problem to one on equivariant birational geometry. For instance, we show that if G and H are both connected, semisimple, simply connected and without compact factors, then G/H is strongly Hardy-Littlewood with respect to some height function. We also show that when H is ``large'' in G and both G and H are connected, reductive and without nontrivial Q-characters, the asymptotic of integral points is the same as the volume asymptotic up to a constant for every equivariant height. Three concrete examples with explicit heights are also provided to illustrate our approach.
In this paper, we consider the problem of visual scanning mechanism underpinning sensorimotor tasks, such as walking and driving, in dynamic environments. We exploit eye tracking data for offering two new cognitive effort measures in visual scanning behavior of virtual driving. By utilizing the retinal flow induced by fixation, two novel measures of cognitive effort are proposed through the importance of grids in the viewing plane and the concept of information quantity, respectively. Psychophysical studies are conducted to reveal the effectiveness of the two proposed measures. Both these two cognitive effort measures have shown their significant correlation with pupil size change. Our results suggest that the quantitative exploitation of eye tracking data provides an effective approach for the evaluation of sensorimotor activities.
Let $X=G/\Gamma$ be the quotient of a semisimple Lie group $G$ by its non-cocompact arithmetic lattice. Let $H$ be a reductive algebraic subgroup of $G$ acting on $X$. We give several equivalent algebraic conditions on $H$ for the existence of a fixed compact set in $X$ intersecting \textit{every} $H$-orbit. This generalizes previous results concerning certain special reductive group action on $X$ in this setting. When $G$ is of real rank one, $\Gamma$ is a non-cocompact lattice of $G$ and $H<G$ is an algebraic group, we also obtain an algebraic condition on $H$ which is equivalent to the return of \textit{every} $H$-orbit to a single compact set in $X$. This complements our results in the case of arithmetic lattice.
A closed horocycle $\mathcal{U}$ on $SL_N(\mathbb{Z}) \backslash SL_N(\mathbb{R})/SO_N({\mathbb{R}})$ has many lifts to the universal cover $SL_N(\mathbb{R})/SO_N({\mathbb{R}})$. Under some conditions on the horocycle, we give a precise asymptotic count of its lifts of bounded distance away from a given base point in the universal cover. This partially generalizes previous work of Mohammadi--Golsefidy.
Let $G/\Gamma$ be the quotient of a semisimple Lie group by an arithmetic lattice. We show that for reductive subgroups $H$ of $G$ that is large enough, the orbits of $H$ on $G/\Gamma$ intersect nontrivially with a fixed compact set. As a consequence, we deduce finiteness result for totally geodesic submanifolds of arithmetic quotients of symmetric spaces that do not admit nontrivial deformation and with bounded volume. Our work generalizes previous work of Tomanov--Weiss and Oh on this topic.
We study an asymptotic formula for counting integral points over an equation defined by a non-degenerated indefinite integral ternary quadratic form $f$ representing a non-zero integer $a$ such that $-a\cdot det(f)$ is square over a number field. In particular, we prove that the finite part of this asymptotic formula is given by the product of local density times $1-p^{-1}$ over all finite primes $p$ over $\Bbb Z$.
Let G be a semisimple linear algebraic group defined over rational numbers, K be a maximal compact subgroup of its real points and {\Gamma} be an arithmetic lattice. One can associate a probability measure {\mu}(H) on {\Gamma}\G for each subgroup H of G defined over Q with no non-trivial rational characters. As G acts on {\Gamma}\G from the right, we can push-forward this measure by elements from G. By pushing down these measures to {\Gamma}\G/K, we call them homogeneous. It is a natural question to ask what are the possible weak-* limits of homogeneous measures. In the non-divergent case this has been answered by Eskin--Mozes--Shah. In the divergent case Daw--Gorodnik--Ullmo prove a refined version in some non-trivial compactifications of {\Gamma}\G/K for H generated by real unipotents. In the present article we build on their work and generalize the theorem to the case of general H with no non-trivial rational characters. Our results rely on (1) a non-divergent criterion on SL_n proved by geometry of numbers and a theorem of Kleinbock--Margulis; (2) relations between partial Borel--Serre compactifications associated with different groups proved by geometric invariant theory and reduction theory.
Let G be a semisimple group over rational numbers and H is a subgroup over rational numbers. Given a representation of G and an integral vector x whose stabilizer is equal to H. In this paper we investigate the asymptotic of integral points on Gx with bounded height. We find its asymptotic up to an implicit constant when H is large in G but we allow the presence of intermediate subgroups. This is achieved by a novel combination of two equidistribution results in two different settings: one is that of Eskin, Mozes and Shah on a Lie group modulo a lattice and the other one is a result of Chamber-Loir and Tschinkel on a smooth projective variety with a normal crossing divisor.
In the present article we study the following problem. Let G be a linear algebraic group over Q, $\Gamma$ be an arithmetic lattice and H be an observable Q-subgroup. There is a H-invariant measure $\mu_H$ supported on the closed submanifold $H\Gamma/\Gamma$. Given a sequence $g_n$ in G we study the limiting behavior of $(g_n)_*\mu_H$. In the non-divergent case we give a rather complete classification. We further supplement this by giving criterion of non-divergence and prove non-divergence for arbitrary sequence $g_n$ for certain H. This work can be viewed as a natural extension of the work of Eskin--Mozes--Shah and Shapira--Zheng.
Given a maximal $\mathbb{Q}$-torus in $SL(N,\mathbb{Q})$, its orbit from identity coset in $SL(N,\mathbb{R})/SL(N,\mathbb{Z})$ naturally carries a possibly infinite Haar measure. We classify all possible limit measures of it when translated by a sequence of elements from $SL(N,\mathbb{R})$. This is a natural extension of Shapira and Zheng's work where only $\mathbb{Q}$-split tori are considered.