arXiv subjects

Xiyu Hu

Publications and source records attributed to Xiyu Hu.

8 recordsLinked to original sources

arXiv2026

Lower bounds for some value sets over finite fields: incidence geometry and Bourgain's group expansion theorem

We develop two transition principles for lower-bounding value sets generated by structured sequences over prime fields. A reciprocal-affine family with $M$ internal transitions and bounded quotient multiplicity has image size $\gg \min{M,p}^{8/15}$. This recovers the factorial-residue bound and yields the same exponent for arithmetic Pochhammer products, Gaussian $q$-factorials, derangement numbers, and the numbers of ordered subsets. A second theorem treats nonzero sequences whose consecutive ratios evolve under a nondegenerate M"obius transformation: their value sets have size $\gg \min{M,p}^{1/2+\eta}$ for an absolute constant $\eta>0$. As consequences, fixed rows of Pascal's triangle and the initial half-blocks of the Catalan and central binomial sequences exceed the square-root scale. The proofs combine transition quotients with, respectively, Cartesian-product point-line incidence geometry and Bourgain's expansion-based incidence theorem in $\mathrm{SL}_2(\mathbb{F}_p)$.

math.NT↗

arXiv2026

Improved Weyl bounds on short intervals

For an integer $d\ge 3$, put $\Delta_d=\min{2^{d-1},d(d-1)}$. Let $a/q$ be reduced, let $P(X)=\frac{a}{q}X^d+\alpha_{d-1}X^{d-1}+\cdots+\alpha_0$, and let $\mathcal{I}$ be an interval of $H\le q$ consecutive integers. We prove $\left|\sum_{n\in\mathcal{I}}e(P(n))\right|\ll_{d,\varepsilon}q^{1/d}H^\varepsilon+H^{1-1/\Delta_d+\varepsilon}$. Consequently, for every prime $p>d$, every degree-$d$ polynomial $P\in\mathbb{F}p[X]$, and every interval $\mathcal{I}$ of $H$ consecutive integers with $p^{1/d}<H<p^{1/(d-1)}$, writing $H^d/p=H^u$, one has $\left|\sum{n\in\mathcal{I}}e_p(P(n))\right|\ll_{d,\varepsilon}H^{1-\min{u/d,1/\Delta_d}+\varepsilon}$. This strictly improves throughout the full natural short-interval window the best generic estimate obtained by combining classical Weyl differencing with the optimal Vinogradov mean value theorem.

math.NT↗

arXiv2026

Lonely Runners over Function Fields: Quantized Phase--Riesz product

Let $C_k(q)$ be the least cardinality of a family of nonzero polynomials over $\mathbb F_q$ whose associated codimension-$k$ partial-circulant kernels cover the full coefficient space. Chow and Rimani'c conjectured that $C_k(q)=1+q+\cdots+q^k$. We disprove the unrestricted conjecture by constructing thirteen monic polynomials over $\mathbb F_2$ whose $k=3$ kernels cover $\mathbb F_2^7$; in particular, $C_3(2)\le 13<15$. For a general covering family of size $N=q^k+S$ and $\mathbb F_q$-linear rank $d$, we prove $S\gg d^{2/3}\left(\frac{\log(2q)}{\log(eNq^k/S)}\right)^{2/3}$. Consequently, for every fixed $k\ge 2$ and all sufficiently large $q$, $C_k(q)\ge q^k+c_kq^{2/3}$. When $k=2$, an integer-multiplicity refinement of the second-moment covering argument yields $\liminf_{q\to\infty}(C_2(q)-q^2)/q\ge \widetilde c_2$, where $\widetilde c_2$ is an explicit one-variable variational constant with numerical value $\widetilde c_2=0.5829944375\ldots$. We also classify triples admitting two independent low-degree polynomial syzygies and prove a conditional packet-free lower bound of size $q^k+(1/2-o(1))q^{k-1}$.

math.CO↗

arXiv2026

Complexity-sensitive additive energy and off-diagonal Young inequalities on bounded-degree algebraic varieties

We develop additive-energy estimates and weighted Young inequalities for finite sets on bounded-degree real algebraic varieties. For an irreducible $m$-dimensional variety $V$, let $\sigma(V)=2m-\dim\overline{V-V}^{\mathrm{Zar}}$ and $\alpha(V)=\max{2,1+\frac{2\sigma(V)}{m}}$. For every $a\in[\alpha(V),3)$ we define a finite-degree translation-partition flag parameter $\Lambda_{a,R}(X;V)$ and prove $E(X)\ll \Lambda_{a,R}(X;V)^{3-a}|X|^{a+\varepsilon}$. This recovers the line-concentration theorem of Jing and Wu for algebraic surfaces in $\mathbb{R}^3$. For codimension-two quadratic threefolds ${(u,Q_1(u),Q_2(u))\in\mathbb{R}^3}\subset\mathbb{R}^5$ with positive-definite $Q_1$ and simple generalized spectrum, we prove the sharp estimate $E(X)\ll_{\varepsilon}|X|^{2+\varepsilon}$ without a flag loss. Hereditary versions of these estimates imply weighted $L^4$ restriction bounds and off-diagonal Young inequalities; at the near-diagonal threshold the sharp region is $1\le p,q\le 2$ and $p^{-1}+q^{-1}\ge 1$. We also prove a sharp turning-complexity extension of the Cushman-Demeter-Wu theorem: $J_3(P)\ll_{\varepsilon}\kappa(P)^2|P|^{3+\varepsilon}$, with matching examples at every power scale.

math.CA↗

arXiv2026

Factorial residues modulo a prime: beyond the square-root bound

For a prime $p$, let $A_p=\{k!\pmod p:1\leq k<p\}$. We prove $|A_p|\gg p^{8/15}$, improving the general lower bound $(\sqrt{2}-o(1))p^{1/2}$. The proof begins with the identity $(n+2)!=(n+1)!+((n+1)!)^2/n!$ in $\mathbb{F}_p$, which produces many incidences for a family of fractional-linear maps. After Cauchy--Schwarz, the transition maps between two members of this family become affine lines, with multiplicity at most two. The Cartesian-product point-line incidence theorem of Stevens and de Zeeuw then yields the exponent $8/15$.

math.NT↗

arXiv2026

Sharp Hausdorff Bounds for the Interior Singular Set of Convex $k$-Hessian Solutions

Let $2\le k\le n$, let $\Omega\subset\mathbb{R}^n$ be open and convex, and let $u$ be a convex viscosity solution of $\sigma_k(D^2u)=1$ in $\Omega$. We prove that the set on which $u$ fails to be locally $C^2$ has vanishing $(n-1)$-dimensional Hausdorff measure. In the intermediate range $3\le k 0$, we obtain Hausdorff bounds for strata defined by the affine dimension of all supporting contact sets. The proof combines a support-dependent Chou--Wang barrier argument, an estimate for the product of the smallest $k$ semiaxes of a John ellipsoid, and Mooney's convex section-covering theorem. As a direct analytical consequence, the full distributional Hessian is absolutely continuous and $u\in W^{2,1}_{\mathrm{loc}}(\Omega)$, yielding a $k$-Hessian counterpart of the $W^{2,1}$ regularity known for singular Monge--Amp\`ere solutions. In a logically separate structural part, we characterize the distinguished number of flat directions, $n-k+1$, by an asymptotic infimum mean-value formula over affine sections, and explain how this mean-value heuristic leads to the supporting-contact geometry used in the proof.

math.AP↗

arXiv2026

From Bounding Boxes to Visual Reasoning: An On-Policy Data Annotation Tool for Vision-Language Models

Vision-language models (VLMs) are rapidly advancing toward sophisticated grounded structured visual reasoning. Training models for such advanced capabilities demands a new genre of data that seamlessly unifies spatial coordinates, open-vocabulary descriptions, structured attributes, and topological relationships into a singular representation. However, existing data annotation tools fundamentally fail to meet these intricate demands, suffering from three systematic bottlenecks: limited expressiveness, severe annotation-training decoupling, and poor data reusability. To bridge this infrastructure gap, we introduce an open-source annotation tool, ScreenAnnotator. First, we define a unified annotation atom schema that binds spatial, semantic, and structural primitives into a single unit. Second, we implement an on-policy annotation loop embedded with a Bayesian Annotation Verifier (BAV). Finally, we design a template-driven multi-task data synthesis process dynamically transforms static atoms into diverse multi-dimensional reasoning tasks, eliminating redundant re-annotation. The on-policy loop drives the annotation accept rate to nearly 100% on flowcharts and 77% on GUI screenshots, while steadily reducing per-image annotation time as labeled data accumulate. In the flowchart scenario, fine-tuning a VLM yields 76.1% average accuracy, which is a 35.1% point absolute gain. Our code is available at: https://github.com/WnQinm/Annotator.

cs.CV↗

arXiv2024

ScreenAgent: A Vision Language Model-driven Computer Control Agent

Existing Large Language Models (LLM) can invoke a variety of tools and APIs to complete complex tasks. The computer, as the most powerful and universal tool, could potentially be controlled directly by a trained LLM agent. Powered by the computer, we can hopefully build a more generalized agent to assist humans in various daily digital works. In this paper, we construct an environment for a Vision Language Model (VLM) agent to interact with a real computer screen. Within this environment, the agent can observe screenshots and manipulate the Graphics User Interface (GUI) by outputting mouse and keyboard actions. We also design an automated control pipeline that includes planning, acting, and reflecting phases, guiding the agent to continuously interact with the environment and complete multi-step tasks. Additionally, we construct the ScreenAgent Dataset, which collects screenshots and action sequences when completing a variety of daily computer tasks. Finally, we trained a model, ScreenAgent, which achieved computer control capabilities comparable to GPT-4V and demonstrated more precise UI positioning capabilities. Our attempts could inspire further research on building a generalist LLM agent. The code is available at \url{https://github.com/niuzaisheng/ScreenAgent}.

cs.HC↗