arXiv subjects

Zhixu Hua

Publications and source records attributed to Zhixu Hua.

3 recordsLinked to original sources

arXiv2026

Cellular Maximal-Density Factoring: Joint Shadings, Stable Refinements, and a Conditional Kakeya Application

Maximal-density factoring must often make one parent object carry several forms of information at once: child mass, two multiplicity levels, a density estimate, and the ability to survive later geometric refinements. These properties are not stable under arbitrary deletion. We introduce a cellular factoring method that places positive child mass on a weighted bipartite graph of labelled parents and half-open spatial cells. After regularizing edge weights and cell degrees, one edge set defines both the child shading and the parent shading. Compatibility and pointwise multiplicity bounds are then exact, while any later selection of whole parent-cell incidences lifts quantitatively back to the children. The combinatorial core does not require Euclidean geometry: it extends to finite measurable atomic partitions with comparable atom measures and admits an explicit finite-iteration loss. We combine this joint construction with a weighted planar incidence estimate, a resolution-limited cellwise angular selection, a labelled slab estimate, and a global synchronization of tubelet and fine-tube mass. The resulting rectangular parent estimate has shading exponent $2-σ$, which is sharp uniformly over the class considered here. With the stated GWZ factoring data and corresponding $K_F(β)$ or $K_{KT}(β)$, we obtain the two small-middle interfaces used in the reduction. For the very-non-sticky branch, assuming both $K_{KT}(β)$ and $K_F(β)$, the stated GWZ Lemma 9.1 hypotheses together with the source-uniformity and refinement interface (I1)--(I4) yield the terminal gain for the represented fine-tube input; this conditional application supplies the corresponding input to the surrounding reduction, while the sharp-convex-parent problem lies outside its scope.

math.CA↗

arXiv2026

Profile-Stable Buffered Multiplicity Factoring in Four Dimensions

Multiplicity factoring is usually formulated for child families at comparable scales. For children of mixed geometry, thickening at the shortest parent scale produces nonuniform inflation ratios, and a single worst-case replacement does not preserve the natural density normalization. We prove a multiplicity-factoring theorem for finite indexed convex parent--child families in $\mathbb{R}^4$ that accommodates arbitrary child shapes, scales, orientations, aspect ratios, and repetitions. The local geometry of each assigned family is encoded by a thickening-weighted Frostman coefficient and a mean-normalized inflation efficiency, both determined by the base family before any shading refinement. The resulting coarse density satisfies $\mathcal{L}^{-A_\varepsilon}(w_1/w_4)^\varepsilon \mathfrak{P}_\varepsilonλ^{K_\varepsilon}$, up to the stated parameter-dependent constant, where $\mathfrak{P}_\varepsilon$ is an explicit profile of the parent loads and local efficiencies. The proof thickens arbitrary measurable shadings, projects along a shortest parent direction, establishes an indexed three-dimensional convex-union estimate in the presence of collisions, and lifts the resulting density response back to four dimensions. A weighted Hölder inequality then assembles the nonuniform parent data, while a common cellular refinement regularizes the fine and coarse multiplicities and yields the stated parentwise multiplicity-product estimate. Under a relative convex Frostman hypothesis, the profile is expressed explicitly in the minimum, mean, and maximum inflation ratios. The comparable-scale regime follows as a specialization after a preliminary load selection. Thus the theorem supplies a structural factoring input for four-dimensional overlap arguments; deriving new Kakeya maximal or Hausdorff-dimension estimates would require additional analytic ingredients.

math.CA↗

arXiv2026

Simple and distinct zeros in a prime-modulus Dirichlet family from near-microscopic to polylogarithmic heights

Fix $η>0$ and $A_0>0$. Let $q$ tend to infinity through odd primes, put $Q=\log q$, and let $T=T(q)$ satisfy \[ \frac{(\log Q)^{1+η}}{Q}\le T\le Q^{A_0}. \] Set $I=(T,2T]$, and sum without weights over the $q-2$ nonprincipal characters modulo $q$. Let $\mathcal N_q$ count nontrivial zeros in $I$ with multiplicity, let $\mathcal N^s_{0,q}$ and $\mathcal N^*_{0,q}$ count simple and distinct zeros on the critical line, and let $\mathcal N_{d,q}$ count all distinct zeros in $I$. Uniformly in this height range, we prove unconditionally \[ \mathcal N_q=\frac{qT\log q}{2π}\{1+o_{η,A_0}(1)\}, \quad \frac{\mathcal N^s_{0,q}}{\mathcal N_q}, \frac{\mathcal N^*_{0,q}}{\mathcal N_q} \ge C_{\mathrm{MT}}-o_{η,A_0}(1), \quad \frac{\mathcal N_{d,q}}{\mathcal N_q}\ge C_d-o_{η,A_0}(1), \] where \[ C_{\mathrm{MT}}=\frac32-\frac1{\sqrt2}\cot\!\left(\frac1{\sqrt2}\right) =0.672500703679\ldots, \qquad C_d=\frac{1+C_{\mathrm{MT}}}{2}=0.836250351839\ldots. \] The proof combines Selberg's family-averaged argument estimate and zero-density deletion with a finite Gevrey Gabor compression of Weil's Hermitian form. Quantitative Fourier--Laplace estimates control exterior zeros down to the lower endpoint, while a local--remote shell decomposition gives uniform control throughout the polylogarithmic upper range. Matrix moment estimates and an inertia-based rank--trace inequality then yield the counting bounds. No form of the generalized Riemann hypothesis is assumed.

math.NT↗