arXiv subjects

Xiang Fang

Publications and source records attributed to Xiang Fang.

At least 19 recordsLinked to original sources

arXiv2026

Positive Toeplitz Operators Between Gaussian Mixed-Norm Fock Spaces

We study positive measure-symbol Toeplitz operators between Gaussian radial--angular mixed-norm Fock spaces throughout the full range of exponents, including infinity. We obtain a complete characterization of boundedness in terms of weighted local masses of the symbol measure, or equivalently in terms of its Gaussian Berezin profile. The criterion separates the transfer of angular integrability from that of radial summability and remains valid in the quasi-Banach and endpoint regimes. We further characterize compactness and obtain the corresponding essential-norm estimates through the asymptotic behavior of the same local-mass and Berezin profiles, and determine the mappings involving the large and little outer-endpoint spaces. We also prove an independent characterization of bounded positive Toeplitz sesquilinear forms. The proofs combine atomic recovery, localized Gaussian matrix estimates, and Rademacher extraction.

math.FA↗

arXiv2026

Toroidal-OAM Hybrid Beam for Divergence-Suppressed OAM Wireless Communication

Orbital angular momentum (OAM) beams offer spatially independent multiplexing in wireless communications, but their donut-shaped intensity diverges with mode order, causing power capture loss at finite apertures and unequal link budgets. We propose a toroidal-OAM hybrid beam that combines the angular confinement and strong longitudinal electric field of toroidal electromagnetic pulses from a coaxial horn with a conventional OAM beam, lowering angular spread and producing non-zero on-axis intensity. A far-field model is built as a weighted superposition of OAM uniform circular array (UCA) radiation, a super-Gaussian toroidal angular envelope, and a zeroth-order Bessel longitudinal field, controlled by three dimensionless parameters (toroidal confinement, coupling strength, longitudinal coefficient) tied to measurable hardware. Numerical results at 3 GHz with a $5λ$ aperture show 12--41% divergence reduction for OAM modes $\ell = 1$--$4$, while modes $\ell \geq 5$ become fully center-peaked. The suppression ratio stays unchanged for fixed $kR$, $κ$, $α$, $β$, and $\ell$. Link evaluation shows 6.4--30.9 dB received SNR gain, over 108% extension in fixed-efficiency link-budget range, and up to five orders of magnitude reduction in multi-mode power imbalance. The central null is filled to 8--100% of the peak; the hybrid beam does not mix OAM modes, with intended-mode power fraction above $0.54$ where the ring is well-defined. OAM mode-division multiplexing capacity at $z=500λ$ shows up to $2.1\times$ ($107\%$) capacity enhancement. Full-wave simulation in ANSYS HFSS corroborates the model, confirming the hybrid launcher preserves the OAM vortex phase while reducing divergence compared with a UCA-only baseline. These findings establish toroidal beam engineering as a practical tool for OAM-based wireless links.

eess.SP↗

arXiv2026

A quantitative form of the planar Falconer distance theorem

We give a quantitative planar form of a multiscale argument for Falconer's distance problem. For every $0<η<1$, we obtain a positive lower bound for the Lebesgue measure of the distance set of a compact subset of the unit ball supporting a probability measure with $ν(B(x,r))\le 2r^{1+η}$. The bound depends only on $η$. For a compact set of diameter $D$ carrying a probability measure with $μ(B(x,r))\le A(r/D)^{1+\varepsilon}$, localization gives $|ΔE|\ge c_{\varepsilon,p}DA^{-p}$ for every fixed $p>1/\varepsilon$. In particular, compact planar sets of Hausdorff dimension greater than one have distance sets of positive Lebesgue measure. The proof develops the multiscale method of the September 2026 manuscript of OpenAI, using one sequence of restricted distance measures for both the mass and Fourier estimates. A reconstruction lemma combines summable total-variation errors with summable squared Fourier norms. No endpoint or sharpness assertion is made.

math.CA↗

arXiv2026

Zepp: Accelerating Distributed MoE Serving under Relaxed Balance Constraints

As Mixture-of-Experts (MoE) models continue to scale, serving them increasingly relies on expert parallelism (EP) across a growing number of devices. Yet skewed expert workloads create imbalance across computation, communication, and memory, making load balancing a central optimization objective in distributed MoE serving. We observe that balance is not free: operations introduced to balance one dimension can themselves be expensive or imbalanced. This motivates us to rethink balance as a constraint rather than an optimization objective. We present Zepp, which directly optimizes the bottleneck communication in distributed MoE serving subject to simplified balance constraints on physical resources, i.e., GPUs and NICs. Zepp progressively optimizes inter-node communication across placement, routing, and execution. It first places expert replicas to reduce token communication under GPU constraints, then reshapes communication flows through split and merge primitives under NIC constraints, and finally partitions and schedules ex- pert computation to overlap the resulting communication. To adapt to dynamic workloads, Zepp jointly coordinates computation, token communication, and expert-weight movement at each iteration. Together, these designs allow Zepp to pursue the most efficient execution rather than a single-dimension balanced one. We implement Zepp and evaluate it against 7 state-of-the-art MoE serving systems, achieving up to 6.68$\times$ MoE layer speedup and a geometric mean speedup of 1.86$\times$ over the fastest competing baseline.

cs.DC↗

arXiv2026

Strict Hausdorff-dimension drop for Quartic Salem Bernoulli convolutions

For every quartic Salem number $β$, we prove that the equal-weight Bernoulli convolution $ν_{β^{-1}}$ has Hausdorff dimension strictly less than one. The main step converts large Fourier coefficients at geometrically separated frequencies into a deficit in the Shannon entropy of a finite convolution. A positive trigonometric product gives the entropy estimate, while the algebraic norm controls the number of distinct atoms in each spatial cell. For the two quartic Salem numbers in $(1,2)$, reciprocal differences at suitable return times produce the required Fourier coefficients. A finite block cover then gives the dimension bound. This proof is independent of numerical computation. Separate exact calculations give $\dimHν_{β_1^{-1}}<1-1.4\cdot10^{-23}$ and $\dimHν_{β_2^{-1}}<1-7.8\cdot10^{-22}$, where $β_1<β_2$ are the two parameters in $(1,2)$. The accompanying verification programs use rational interval arithmetic and the Python standard library.

math.DS↗

arXiv2026

Strict correlation-dimension drop for quartic Salem Bernoulli convolutions

For every quartic Salem number $β\in(1,2)$, we prove that the Bernoulli convolution $ν_{β^{-1}}$ with equal weights has correlation dimension strictly less than one. In particular, neither of the two corresponding measures has an $L^2$ density. We relate correlation dimension to exact collision probabilities and express these probabilities as positive integrals over a four-dimensional torus associated with the defining polynomial. The unit-circle conjugates produce a persistent oscillatory term; using carefully chosen return times of the corresponding rotation, we take differences that make this term small while controlling the length and cost of the resulting configurations. Many well-separated placements of these configurations then give a collision probability above the critical scale, which yields the strict dimension drop. The qualitative argument is independent of the computer-assisted estimates. Separate computer-assisted arguments give \[ 0.99999<D_2(ν_{β_1^{-1}})<1-10^{-65}, \qquad 0.99999<D_2(ν_{β_2^{-1}})<1-10^{-75}, \] where $β_1<β_2$ are the two quartic Salem numbers in $(1,2)$.

math.DS↗

arXiv2026

Fourier asymptotics of critical Gaussian multiplicative chaos on the circle

We study the high-frequency Fourier coefficients $C_n$ of critical Gaussian multiplicative chaos $M$ on the circle, for logarithmic covariances with an arbitrary admissible smooth, possibly nonstationary remainder. For every fixed finite set of integer offsets, the corresponding vector of neighboring coefficients, multiplied by $\sqrt{\log n}$, converges stably relative to the full Gaussian field without additional centering. Conditionally on that field, the limit consists of Fourier coefficients of one symmetric complex stable random measure of index one with control proportional to $M$. For every $s>1$, the normalized modulated measures also converge stably in $H^{-s}(\mathbb{T})$ to this conditional stable noise. Unconditionally, the same joint approximation remains valid for offsets of size at most $(\log\log n)^{1/64}$ for every fixed admissible covariance. The scalar limit is an isotropic Cauchy mixture; the joint limiting law retains the spatial distribution of $M$, not only its total mass. We also prove an almost-sure integral test along the entire integer frequency sequence. For a class of regular deterministic gauges $H$, the limsup of $\sqrt{\log|n|}\,|C_n|/H(\log\log|n|)$ is zero or infinity according as $\int^\infty dx/H(x)$ converges or diverges. The proof combines index-one compensation with relative excursion estimates and transfers these conclusions from the canonical model by smooth covariance comparison.

math.PR↗

arXiv2026

Beyond Parallel Blindness: Information Floors and Model Gaps in Block Drafting

Block drafters propose several tokens in one forward pass, before earlier target tokens are realised. Their rejection mixes two losses: missing within-block path information and imperfect modelling of observable information. Accepted length cannot distinguish them. We separate the two with an information floor, the minimum expected rejection at a specified conditioning order; rejection above this floor is the model gap. Measurements across four domains and four open-weight targets, plus floor estimates on a frontier API target, yield three findings. First, the all-parallel floor reaches $0.286$ at the final slot on Qwen3-4B, limiting even the best proposal to $71\%$ per-slot acceptance. Second, one realised token removes $86$--$100\%$ of this floor, a locality also recovered by an independent mutual-information analysis. Third, current drafters remain far above their floors: the final-slot model gap accounts for $43$--$64\%$ of DFlash rejection and $85$--$92\%$ of DSpark's oracle-conditioned rejection. Guided by this separation of short-range conditioning from proposal quality, we replace a context-independent predecessor correction with a prefix-attention head that reads committed context conditional on the predecessor. With supporting backbone components, the resulting Qwen3-4B drafter improves mean serving accepted length by $3.93\%$ over the released DSpark checkpoint across nine tasks, without increasing the conditioning order. Code and data: https://github.com/tie-pilot-qxw/specfloor; drafter checkpoints: https://huggingface.co/TIE-Pilot/dspark-attnconv-block7-qwen3-4b.

cs.LG↗

arXiv2026

Sampling and Interpolation in Gaussian Mixed-Norm Fock Spaces

We establish sharp sampling and interpolation criteria for the radial--angular Gaussian mixed-norm Fock spaces $\mathcal F_α^{p,q}$, where $α>0$ and $0 \fracαπ$, together with relative separation when $p<\infty$; no relative-separation condition is required when $p=\infty$. Here $D_{\mathrm{sep}}^{-}$ is the supremum of the lower Beurling densities over separated subsets of $Λ$. Interpolation is equivalent to separation and $D^{+}(Λ)<\fracαπ$. The same criterion characterizes interpolation for the little endpoint $f_α^{p,\infty}$, and in both settings the normalized restriction map admits a bounded linear right inverse. The sufficiency arguments are based on rapidly localized Hilbert dual and Lagrange atoms together with a weighted localized synthesis theorem on the full mixed-norm scale. For necessity, we prove that lower stability of a rapidly localized matrix on a weighted annular mixed sequence space implies lower stability on $\ell^2$. The proof first passes to $\ell^\infty$ by translated polynomial cutoffs and a commutator estimate, and then uses the $p$-independence theorem for the Sjöstrand class. Applied through a fixed Hilbert sampling lattice, this reduces the strict density conditions to the classical Hilbert Fock sampling and interpolation theorems.

math.CV↗

arXiv2026

Sharp Gaussian Mixed-Norm Theory for Fock Spaces

We develop a structural theory of the Gaussian mixed-norm Fock spaces $\mathcal F_α^{p,q}$, where $α>0$ and $0<p,q\leq\infty$, which separate angular $L^p$-means from radial Gaussian $L^q$-summability. We first prove the sharp circle-mean estimate $M_p(f,r)\leq C_{α,q}(1+r)^{-\frac{1}{q}} e^{\fracα{2}r^2}\lVert f\rVert_{\mathcal F_α^{p,q}}$, $r\geq0$, where $M_p(f,r)$ denotes the angular $L^p$-mean of $f$ on the circle $\lvert z\rvert=r$ with the convention $\frac{1}{\infty}=0$, and show that the exponent $-\frac{1}{q}$ is optimal. The proof is based on a local mass principle for convex Laplace-type exponents, which also yields an annular discretization of the mixed Gaussian norm. Using this discretization, we characterize the continuous embeddings between mixed-norm Fock spaces and obtain an atomic decomposition in terms of normalized Fock kernels, with coefficients belonging to a corresponding annular mixed sequence space. We further characterize Carleson and vanishing Carleson measures and identify the continuous duals in the Banach, quasi-Banach, and endpoint regimes. For $p<1$, the atomic decomposition relies on a quantitative re-centering theorem that controls expansions of normalized Fock kernels under perturbations of their centers. This theorem is of independent interest even for the Hilbert Fock space $\mathcal F_α^{2}$.

math.CV↗

arXiv2026

Canonical analytic realizations of hyperbolic determinantal processes

Krishnapur asked whether the invariant hyperbolic determinantal point processes on the disk admit a random analytic zero-set interpretation at noninteger parameters. We construct such a realization for every positive real parameter as the full compact-open limit in distribution of normalized finite Blaschke products. The zeros determine the modulus and normalized analytic shape, leaving one independent uniform phase. We prove exact Möbius covariance and classify all realizations with this covariance and square-integrable logarithmic modulus at the origin: they are precisely independent positive random multiples of the canonical function. Within this covariant class, matching the canonical logarithmic mean and variance uniquely determines the canonical function law. The family is weakly continuous in the parameter and agrees at positive integers with determinants of matrix-valued Gaussian power series. An explicit Barnes $G$-function Mellin transform determines the basepoint normalization.

math.PR↗

arXiv2026

The Local Embedding Problem for Hardy Spaces of Dirichlet Series

We solve the local embedding problem for Hardy spaces of Dirichlet series, which is a dimension-free trace problem asking whether the global $\mathscr{H}^p$-norm controls local $L^p$-mass on the critical line $\operatorname{Re}s=1/2$. More precisely, for every $2<p<\infty$, there exists a constant $C_p<\infty$ such that every Dirichlet polynomial $P$ satisfies $$ \sup_{θ\in\mathbb{R}}\int_θ^{θ+1}\left|P\left(\frac12+it\right)\right|^p\,\mathrm{d}t\le C_p\left\lVert P\right\rVert_{\mathscr{H}^p}^{p}, $$ with $C_p$ independent of the number and choice of prime variables on which $P$ depends. Before the present work, the embedding was known at $p=2$ and, by taking integer powers, at the even exponents $p=2k$; it had been conjectured that these exhaust the finite positive cases above $2$. Together with the known failure for $0<p<2$, our theorem gives the sharp finite-exponent classification: the local embedding property holds exactly for $p\ge2$. Thus, the true threshold is $p=2$, rather than even integrality. The proof passes to the dual exponent $q=p/(p-1)\in(1,2)$, where an exact frequency decomposition isolates a single resonant Euler-product term. A covariance-preserving replacement of the shared prime factors reduces the resulting fractional-moment estimate to a log-correlated Gaussian field, and a critical branching-random-walk bound supplies the required multiscale decay. A finite-cyclic square-function estimate assembles the resonant scales, and Hardy-quotient duality converts the resulting vector-valued bound into the critical-line trace. For $1\le p<\infty$, known equivalences give the same sharp threshold in several classical problems, including the conformally invariant half-plane embedding, the reverse local Carleson-measure transfer, and boundedness of all characteristic-zero Gordon--Hedenmalm composition operators.

math.FA↗

arXiv2026

Forbid Your Attention: Fooling Multimodal Large Language Models by Selectively Removing Intrinsic Focus in Spectral Domain

Multimodal large language models (MLLMs) have extended the capability of large language models (LLMs) to process more contextual multimodal information, showing remarkable progress in diverse realistic multimodal applications. Despite their strong perception and reasoning abilities, recent studies reveal that MLLMs remain highly vulnerable to adversarial inputs, especially those targeting visual components. However, existing attacks mainly focus on global perturbations, lacking an understanding of how MLLMs internally interpret visual structures. In this paper, we make the attempt to investigate the intrinsic focus of MLLMs in the frequency domain and discover that their predictions are particularly sensitive to phase information, which encodes essential structural and semantic cues. Based on this observation, we propose a novel phase-aware adversarial attack framework that explicitly restricts adversarial perturbations to structure-relevant phase regions to suppress the MLLMs' focus for effective and imperceptible attacks. To further amplify the structural influence, we also introduce an auxiliary adversarial prompt learning module to guide multimodal misalignment around phase-sensitive regions, misleading the MLLM's attention toward targeted structural patterns. Extensive experiments on multiple representative MLLM models and datasets demonstrate the superior effectiveness of our method compared to existing attacks.

cs.CV↗

arXiv2026

Critical Gaussian Multiplicative Chaos on the Circle Is Rajchman

We prove that the Fourier coefficients of the canonical critical Gaussian multiplicative chaos on the circle vanish almost surely at infinity. More precisely, let $M_ϕ^{\mathrm{crit}}$ be the canonical critical chaos associated with the centered circle field $ϕ$ of covariance $\mathbb{E}[ϕ(θ)ϕ(θ')] = \log\frac{1}{\lvert e^{iθ}-e^{iθ'}\rvert}$. Then, almost surely, $\widehat{M_ϕ^{\mathrm{crit}}}(n)\longrightarrow0$ as $\lvert n\rvert\to\infty$. This resolves the almost-sure critical Rajchman problem for the canonical circle field. Since critical chaos has Fourier dimension zero almost surely, no positive polynomial Fourier-decay rate can hold; the theorem therefore exhibits qualitative Fourier cancellation beyond the regime of positive Fourier dimension. The proof addresses two coupled difficulties: the heavy, nonuniform cell masses of critical chaos and the need to control exponentially many frequencies in each dyadic annulus. For an auxiliary periodized compact-range star-scale field, a derivative-rooted Bessel regression yields weighted small-cell summability and moving-tail control of exceptional large cells. After conditioning at a coarse scale below the Fourier scale, finite-range independence and conditional Bernstein concentration reduce uniform control of the terminal Fourier coefficients over each dyadic annulus to a spatial-variation estimate for a coarse predictable measure. A smooth positive-definite covariance correction and critical-chaos uniqueness then transfer the Rajchman property to the canonical critical chaos of the exact circle field.

math.PR↗

arXiv2026

LightStim: A Framework for QEC Protocol Evaluation and Prototyping with Automated DEM Construction

Fault-tolerant quantum computing increasingly demands rigorous, circuit-level evaluation of diverse quantum error correction (QEC) protocols and efficient prototyping of new ones. Such evaluation requires both the physical circuit and its Detector Error Model (DEM) to estimate end-to-end logical error rates. However, DEM construction today is performed by manual annotation, a tedious and error-prone process that effectively limits evaluation to simple memory experiments. We present LightStim, a framework that automates DEM construction concurrently with circuit compilation by maintaining a Pauli tableau augmented with measurement records, with no protocol-specific input required. We benchmark LightStim across protocols from memory experiments to end-to-end distillation circuits; cross-validation against public implementations confirms exact detector and observable counts and consistent logical error rates. Additionally, we demonstrate a novel heterogeneous cross-code lattice surgery design between surface and punctured quantum Reed-Muller codes. These capabilities together make LightStim a unified infrastructure for systematic QEC protocol evaluation and exploration. LightStim is open-sourced at https://github.com/QuTone/LightStim.

quant-ph↗

arXiv2026

Critical Norm Profiles for Finite-Prime Composition Operators on the Hardy Space of Dirichlet Series

We identify the critical boundary operator-norm profile of finite-prime composition operators on the Hardy--Hilbert space $\mathcal H^2$ of Dirichlet series. For \[ φ_{δ,\boldsymbolρ}(s) = \frac12+δ+ δ\sum_{j=1}^dρ_jp_j^{-s}, \qquad \boldsymbolρ\in B_d, \] the renormalized positive coefficient operators converge uniformly in operator norm, with $O(δ)$ error, to an explicit multivariate weighted Hankel operator $\mathcal H_{\boldsymbolρ}$; consequently, \[ 2δ\|C_{φ_{δ,\boldsymbolρ}}\|^2 = \|\mathcal H_{\boldsymbolρ}\| + O(δ) \] uniformly over $B_d$. We show that the limiting operator admits the total-degree reduction \[ \mathcal H_{\boldsymbolρ} \simeq D_{\boldsymbolρ} H_{R_{\boldsymbolρ}/2} D_{\boldsymbolρ}\oplus\mathbf{0}, \] where the diagonal factors are convolution-collision norms of the normalized prime weights. This structure, together with the affine comparison principle of Brevig and Perfekt, yields an explicit concentration inequality for $\|\mathcal H_{\boldsymbolρ}\|$, identifies the one-prime configurations as the exact equality cases in the limiting norm estimate, and gives a quantitative deficit away from them. For fixed $σ>\frac12$, we also obtain a second-order expansion of the squared norm and fully finite-dimensional approximations with explicit total-degree and Dirichlet-sum truncation errors. Together, these results show that a single coefficient-operator structure governs the singular boundary profile, the fixed-$σ$ perturbative regime, and certified finite-dimensional approximation.

math.FA↗

arXiv2026

Hardy-Szegő Point Processes: Large Deviations and Strong Szegő Asymptotics

We study exponential-scale fluctuations of the Hardy-Szegő zero process, investigated by Peres and Virág in the disk setting, in its upper-half-plane realization, which reveals a different probabilistic geometry. This conformally invariant determinantal zero process is equivalent to its disk realization, but the upper-half-plane coordinates make real-translation invariance explicit and single out long horizontal windows as natural observables. For the vertical window from height one to height $a>1$, let $N_a(L)$ denote the number of zeros in the corresponding horizontal window of length $L$. We identify the limiting scaled log-moment generating function explicitly. As a consequence, we prove a large deviations principle for $N_a(L)/L$, with rate function given by the Legendre transform of this limit. We also prove a strong Szegő expansion for the log-moment generating function, including an explicit order-one correction, locally uniformly in the natural complex strip. The proof uses the planar determinantal structure before projection, reduces the problem to a one-dimensional Fredholm determinant, and combines fixed-power trace asymptotics with a Wiener-Hopf comparison.

math.PR↗

arXiv2026

Two Regularity Problems on Analytic Tent Spaces

We study two regularity problems on Hardy-type analytic tent spaces $\mathcal{AT}^p_{q,α}$ on the unit disk: fractional integration and randomization of Taylor coefficients. For fractional integration, we characterize completely the boundedness and compactness of the Hadamard, Flett, and Riemann--Liouville operators between analytic tent spaces, and obtain parallel results for analytic Triebel--Lizorkin spaces. In particular, the case $t=0$ yields a complete solution to the corresponding embedding problem for analytic tent spaces. For randomization, we characterize completely when the random Taylor series $\mathcal{R}f$ belongs almost surely to an analytic tent space whenever $f\in \mathcal{AT}^p_{q,α}$, and we also obtain the Triebel--Lizorkin counterpart. As part of the proof, we identify the random symbol space associated with $\mathcal{AT}^p_{q,α}$ and solve the embedding problem from analytic tent spaces into mixed norm spaces. These results extend classical theorems of Hardy--Littlewood and Littlewood, as well as their later analogues for Bergman and mixed norm spaces.

math.FA↗