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Zuo Lin

Publications and source records attributed to Zuo Lin.

7 recordsLinked to original sources
arXiv2026

Translation Surfaces arising from Right Regular Prisms

We study flat metrics arising from right regular $n$-prisms by viewing them as $n$-differentials and analyzing their associated unfoldings. We show that the unfolding of a right regular $n$-prism is never a lattice surface unless $n=4$, in contrast with the case of Platonic solids. Despite this, we prove that these surfaces admit translation coverings to hyperelliptic surfaces, allowing us to determine their $\mathrm{GL}(2,\mathbb{R})$-orbit closures using the classification of hyperelliptic components of strata. As a consequence, we obtain exact quadratic asymptotics for a certain average of the number of saddle connections on the base surfaces, their unfoldings, and the original prisms, including their Siegel--Veech constants. This provides a natural infinite family of non-lattice surfaces for which orbit closures and counting problems can be computed explicitly.

math.GT↗
arXiv2025

Polynomially effective equidistribution for certain unipotent subgroups in quotients of perfect Lie groups

We prove an effective equidistribution theorem for orbits of certain unipotent subgroups in arithmetic quotients of perfect Lie groups with a polynomial error term. Even for semisimple quotients, our result provides the first infinite family of examples where effective equidistribution with polynomial error rate is obtained for non-horospherical unipotent subgroups. The proof is based on the spectral gap of the ambient space, an effective closing lemma, Bourgain's discretized projection theorem, and a sub-modularity inequality in irreducible representation. The sub-modularity inequality is crucial to our proof and is of independent interest. As applications, we obtain effective estimates on distribution of lattice orbits on homogeneous spaces, as well as an effective version of the Oppenheim conjecture for indefinite quadratic forms with a polynomial error rate in all dimension $d \geq 3$.

math.DS↗
arXiv2025

Quadratic forms of signature $(2, 2)$ or $(3, 1)$ I: effective equidistribution in quotients of $\mathrm{SL}_4(\mathbb{R})$

We prove an effective equidistribution theorem for orbits of horospherical subgroups of $\mathrm{SO}(2, 2)$ and $\mathrm{SO}(3, 1)$ in quotients of $\mathrm{SL}_4(\mathbb{R})$ with a polynomial error term. In a forthcoming paper, we will use this theorem to prove an effective version of the Oppenheim conjecture for indefinite quadratic forms of signature $(2, 2)$ or $(3, 1)$ with a polynomial error rate.

math.DS↗
arXiv2024

Restricted projections and Fourier decoupling in $\mathbb{Q}_p^n$

We prove a restricted projection theorem for Borel subsets of $\mathbb{Q}_p^n$ in the regime $p>n$. This generalizes results of Gan-Guo-Wang in the real setting. Our result is effective in the sense that explicit constants are obtained for various covering numbers. Along the way, we prove a fully explicit Fourier decoupling theorem for the moment curve in $p$-adic Cartesian space.

math.CA↗
arXiv2024

Polynomial effective density in quotient of $\mathrm{SL}_2(\mathbb{Q}_p) \times \mathrm{SL}_2(\mathbb{Q}_p)$

We prove an effective density theorem with polynomial error rate for orbits of upper triangular subgroup of $\mathrm{SL}_2(\mathbb{Q}_p)$ in $\mathrm{SL}_2(\mathbb{Q}_p) \times \mathrm{SL}_2(\mathbb{Q}_p)$ for prime number $p > 3$. The proof is based on the use of Margulis function, a restricted projection theorem on $\mathbb{Q}_p^3$, and spectral gap of the ambient space.

math.DS↗
arXiv2024

Finitary estimates for the distribution of lattice orbits in homogeneous spaces I: Riemannian metric

Let $H < G$ both be noncompact connected semisimple real algebraic groups where the former is maximal proper and $\Gamma < G$ be a lattice. Building on the work of Gorodnik-Weiss, we refine their techniques and obtain effective results. More precisely, we prove effective convergence of the distribution of dense $\Gamma$-orbits in $G/H$ to some limiting density on $G/H$ assuming effective equidistribution of regions of maximal horospherical orbits under one-parameter diagonal flows inside a dense $H$-orbit in $\Gamma \backslash G$. The significance of the effectivized argument is due to the recent effective equidistribution results of Lindenstrauss-Mohammadi-Wang for $\Delta(\operatorname{SL}_2(\mathbb R)) < \operatorname{SL}_2(\mathbb R) \times \operatorname{SL}_2(\mathbb R)$ and $\operatorname{SL}_2(\mathbb R) < \operatorname{SL}_2(\mathbb C)$ and arithmetic lattices $\Gamma$, and future generalizations in that direction.

math.DS↗