Jingwei Guo, Tao Jiang
We study the lattice point problem associated with a special class of high-dimensional finite type domains via estimating the Fourier transforms of corresponding indicator functions.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
继续探索
We study the Fourier transforms of indicator functions of some special high-dimensional finite type domains and obtain estimates of the associated lattice point discrepancy.
We study the variance of the random variable that counts the number of lattice points in some shells generated by a special class of finite type domains in $\mathbb R^d$. The proof relies on estimates of the Fourier transform of indicator functions of convex domains.
Theta invariants and Lattice-Point Counting in Normed $\mathbb{Z}$-Modules
本文运用欧几里得格理论研究有限秩赋范$\text{Z}$-模，基于θ级数解析性质建立其格点计数函数的不等式及相关结果，拓展了数论与几何领域的相关理论。
We count primitive lattices of rank $d$ inside $\mathbb{Z}^{n}$ as their covolume tends to infinity, with respect to certain parameters of such lattices. These parameters include, for example, the subsapce that a lattice spans, namely its projection to the Grassmannian; its homothety class; and its equivalence class modulo rescaling and rotation, often referred to as a shape. We add to a prior work of Schmidt by allowing sets in the spaces of parameters that are general enough to conclude joint equidistribution of these parameters. In addition to the primitive $d$-lattices themselves, we also …
We determine the normalizer in $SL_{2}(\mathbb{R})$ of several families of congruence subgroups of $SL_{2}(\mathbb{Z})$. In addition, we show how these tools can be used to evaluate the groups of automorphisms and the discriminant kernels of a large family of isotropic lattices of signature (2,1).
This paper is a continuation of the author's previous work, where we studied the variation of the number of isomorphic classes of $G_{\mathbb{Q}}$-stable lattices in p-adic families of residually reducible ordinary Galois representations. In this paper, we study the number of isomorphic classes of $G_{\mathbb{Q}}$-stable free lattices in a residually reducible Hida deformation and the variation of their related two-variable algebraic $p$-adic $L$-functions. This result gives an answer of a question asked by Ochiai.
The density Hypothesis for irreducible lattices in $\mathrm{PSL}_2(\mathbb{R})^d$
本文证明了 $\mathrm{PSL}_2(\mathbb{R})^d$ 中不可约均匀格子的同余子群的密度假设，将球面情形推广至非温非球面表示，并给出关于水平和谱参数的一致界。
Let $K\subseteq\mathbb{R}^2$ be a compact set such that $K+\mathbb{Z}^2=\mathbb{R}^2$. We prove, via Algebraic Topology, that the integer points of the difference set of $K$, $(K-K)\cap\mathbb{Z}^2$, is not contained on the coordinate axes, $\mathbb{Z}\times\{0\}\cup\mathbb{Z}\times\{0\}$. This result gives a negative answer to a question posed by P. Hegarty and M. Nathanson on relatively prime lattice points.
An improved upper bound on the covering radius of the logarithmic lattice of $\mathbb{Q}(ζ_n)$
本文研究分圆数域$\u200b\mathbb{Q}(ζ_n)$对应的对数格，改进了de Araujo 2024年工作中的上界及相关引理，给出用n和其不同素因子个数表示的覆盖半径上界，并证明渐近意义下无法进一步改进。
Let $B$ be a Borel set in $\mathbb E^{d}$ with volume $V(B)=\infty$. It is shown that almost all lattices $L$ in $\mathbb E^{d}$ contain infinitely many pairwise disjoint $d$-tuples, that is sets of $d$ linearly independent points in $B$. A consequence of this result is the following: let $S$ be a star body in $\mathbb E^{d}$ with $V(S)=\infty$. Then for almost all lattices $L$ in $\mathbb E^{d}$ the successive minima $λ_{1}(S,L),..., λ_{d}(S,L)$ of $S$ with respect to $L$ are 0. A corresponding result holds for most lattices in the Baire category sense. A tool for the latter result is the sem…
Polarized superspecial abelian varieties over $\mathbb{F}_p$ via hermitian lattices
本文通过希格斯晶格研究 $\mathbb{F}_p$ 上极化超特殊阿贝尔簇的同构类，确定其非空条件并分类其族。
Evaluating lattice sums via telescoping on $SL_+(2,\mathbb Z)$: a short proof of $\sum \frac{1}{|x|^2|y|^2|x+y|^2}=\fracπ{4}$ and Zagier's identity
本文通过在SL_+(2,Z)上引入telescoping方法，证明了格点求和∑1/(|x|²|y|²|x+y|²)=π/4，并给出了Zagier恒等式的简短证明。
Let $L=diag(1,1,\ldots,1,-1)$ and $M=diag(1,1,\ldots,1,-2)$ be the lattices of signature $(n,1)$. We consider the groups $Γ=SU(L,\mathcal{O}_K)$ and $Γ'=SU(M,\mathcal{O}_K)$ for an imaginary quadratic field $K=\mathbb{Q}(\sqrt{-d})$ of discriminant $D$ and it's ring of integers $\mathcal{O}_K$, $d$ odd and square free. We compute the Hirzebruch-Mumford volume of the factor spaces $\mathbb{B}^n/Γ$ and $\mathbb{B}^n/Γ'$. The result for the factor space $\mathbb{B}^n/Γ$ is due to Zeltinger, but as we're using it to prove the result for $\mathbb{B}^n/Γ'$ and it is hard to find his article, we prov…
It is a well-known result that the proportion of lattice points visible from the origin is given by $\frac{1}{ζ(2)}$, where $ζ(s)=\sum_{n=1}^\infty\frac{1}{n^s}$ denotes the Riemann zeta function. Goins, Harris, Kubik and Mbirika, generalized the notion of lattice point visibility by saying that for a fixed $b\in\mathbb{N}$, a lattice point $(r,s)\in\mathbb{N}^2$ is $b$-visible from the origin if no other lattice point lies on the graph of a function $f(x)=mx^b$, for some $m\in\mathbb{Q}$, between the origin and $(r,s)$. In their analysis they establish that for a fixed $b\in\mathbb{N}$, the p…
1-Point Functions for $\mathbb{Z}_2$-Orbifolds of Lattice VOAs
本文计算了格子顶点算子代数$\mathbb{Z}_2$对称破缺下所有态的一点关联函数，揭示了对称破缺对代数结构的影响。
We consider the ensemble of random Gaussian Laplace eigenfunctions on $\mathbb{T}^3=\mathbb{R}^3/\mathbb{Z}^3$ (`$3d$ arithmetic random waves'), and study the distribution of their nodal surface area. The expected area is proportional to the square root of the eigenvalue, or `energy', of the eigenfunction. We show that the nodal area variance obeys an asymptotic law. The resulting asymptotic formula is closely related to the angular distribution and correlations of lattice points lying on spheres.
$\mathbb{Z}_p$-lattices in semistable Galois representations
该研究证明对数棱柱F-晶体范畴与半稳定Galois表示中的ℤₚ-格范畴等价，并借助Breuil–Kisin模的对数版本用线性代数数据描述这类伽罗瓦表示，为数论中伽罗瓦表示研究提供新的范畴等价工具。
Fix an positive integer $n$. Let $K\subseteq\mathbb{R}^n$ be a compact set such that $K+\mathbb{Z}^n=\mathbb{R}^n$. We prove, via Algebraic Topology, that the integer points of the difference set of $K$, $(K-K)\cap\mathbb{Z}^n$, is not contained on the coordinate axes, $\mathbb{Z}\times\{0\}\times\ldots\times\{0\}\cup\{0\}\times\mathbb{Z}\times\ldots\times\{0\}\cup\ldots\cup\{0\}\times\{0\}\times\ldots\times\mathbb{Z}$. This result gives a negative answer to a question posed by P. Hegarty and M. Nathanson on relatively prime lattice points.
We study an optimal stretching problem for certain convex domain in $\mathbb{R}^d$ ($d\geq 3$) whose boundary has points of vanishing Gaussian curvature. We prove that the optimal domain which contains the most positive (or least nonnegative) lattice points is asymptotically balanced. This type of problem has its origin in the "eigenvalue optimization among rectangles" problem in spectral geometry. Our proof relies on two-term bounds for lattice counting for general convex domains in $\mathbb{R}^d$ and an explicit estimate of the Fourier transform of the characteristic function associated with…
For any finite set $M\subset {\mathbb Z}_{\geq 1}$ of positive integers, there is up to isomorphism a unique ${\mathbb Z}$-lattice $H_M$ with a cyclic automorphism $h_M:H_M\to H_M$ whose eigenvalues are the unit roots with orders in $M$ and have multiplicity 1. The paper studies the automorphisms of the pair $(H_M,h_M)$ which have eigenvalues in $S^1$. The main result are necessary and sufficient conditions on the set $M$ such that the only such automorphisms are $\pm h_M^k,k\in{\mathbb Z}$. The proof uses resultants and cyclotomic polynomials. It is elementary, but involved. Special cases of …
In this review, we count and classify certain sublattices of a given lattice, as motivated by crystallography. We use methods from algebra and algebraic number theory to find and enumerate the sublattices according to their index. In addition, we use tools from analytic number theory to determine the asymptotic behaviour of the corresponding counting functions. Our main focus lies on similar sublattices and coincidence site lattices, the latter playing an important role in crystallography. As many results are algebraic in nature, we also generalise them to $\mathbb{Z}$-modules embedded in $\ma…
There is a well-known asymptotic formula, due to W. M. Schmidt (1968) for the number of full-rank integer lattices of index at most $V$ in $\mathbb{Z}^n$. This set of lattices $L$ can naturally be partitioned with respect to the factor group $\mathbb{Z}^n/L$. Accordingly, we count the number of full-rank integer lattices $L \subseteq \mathbb{Z}^n$ such that $\mathbb{Z}^n/L$ is cyclic and of order at most $V$, and deduce that these co-cyclic lattices are dominant among all integer lattices: their natural density is $\left(ζ(6) \prod_{k=4}^n ζ(k)\right)^{-1} \approx 85\%$. The problem is motivat…
New $s$-extremal extremal unimodular lattices in dimensions $38$, $40$, $42$ and $44$ are constructed from self-dual codes over $\mathbb{F}_5$ by Construction A. In the process of constructing these codes, we obtain a self-dual $[44,22,14]$ code over $\mathbb{F}_5$. In addition, the code implies a $[43,22,13]$ code over $\mathbb{F}_5$. These codes have larger minimum weights than the previously known $[44,22]$ codes and $[43,22]$ codes, respectively.
Let $L =\sqrt{\frac{1}{\Im(z)}}\Big({\mathbb Z}\oplus z{\mathbb Z}\Big)$ where $z \in \mathbb{H}=\{z= x+ i y\;\hbox{or}\;(x,y)\in\mathbb{C}: y>0\}$ be the two dimensional lattices with unit density. Assuming that $α\geq1$, we prove that \begin{equation}\aligned\nonumber \min_{L}\sum_{\mathbb{P}\in L, |L|=1}|\mathbb{P}|^2 e^{- πα|\mathbb{P}|^2} \endaligned\end{equation} is achieved at hexagonal lattice. More generally we prove that for $α\geq 1$ \begin{equation}\aligned\nonumber \min_{L}\sum_{\mathbb{P}\in L, |L|=1}(|\mathbb{P}|^2-\frac{b}α) e^{- πα|\mathbb{P}|^2} \endaligned\end{equation} is a…
The distribution of intersections in $\mathrm{SL}(3, \mathbb{Z}) \backslash \mathrm{SL}(3, \mathbb{R})$ and lattices related to roots of cubic congruences
研究正对角子群闭轨与极大抛物子群平移的交点分布，证明交点在子群中联合等分布，并导出三次同余根对确定的仿射格点与理想联合等分布。
After recalling some basic notions of quantum statistical mechanics, we explain the Bost-Connes system that relates the structure of the maximal abelian extension of $\mathbb{Q}$ to the space of \kms states of a \cs-dynamical system. Afterwards, we study briefly the Connes-Marcolli $\text{GL}_2$-system as a generalization of the former system.
A lattice point counting approach for the study of the number of self-avoiding walks on $\mathbb{Z}^{d}$
该研究通过将正方形格自回避行走计数简化为多维域整点计数，得到其长度为$n$的渐近估计，并为任意$d\geq 2$维$\mathbb{Z}^{d}$格自回避行走数量性质研究提供统一框架。
Sharp o-minimality and lattice point counting
本文在sharp o-极小结构与$\mathbb{R}_{\exp}$的语境下，分别给出了Barroero和Widmer关于o-极小可定义族纤维格点计数精确估计结果的有效版本。
Let $X$ denote the non-compact globally Hermitian symmetric space of type $DIII$, namely, $\text{SO}(n,\mathbb{H})/\text{U}(n)$. Let $Λ$ be a uniform torsionless lattice in $\text{SO}(n,\mathbb{H})$. In this note we construct certain complex analytic submanifolds in the locally symmetric space $X_Γ:=Γ\backslash \text{SO}(n,\mathbb{H})/\text{U}(n)$ for certain finite index sub lattices $Γ\subset Λ$ and show that their dual cohomology classes in $H^*(X_Γ;\mathbb{C})$ are not in the image of the Matsushima homomorphism $H^*(X_u; \mathbb{C})\to H^*(X_Γ;\mathbb{C})$, where $X_u=\text{SO}(2n)/\text{…
Semigroups from full lattices in commutative ${\mathbb Q}$-algebras
本文研究了交换有理代数中满格构成的半群及其商半群，扩展了乔丹-扎森豪斯定理，并探讨了其在正交群共轭类中的应用。
Consider an arbitrary $n$-dimensional lattice $Λ$ such that $\mathbb{Z}^n \subset Λ\subset \mathbb{Q}^n$. Such lattices are called {\it rational} and can always be obtained by adding $m \le n$ rational vectors to $\mathbb{Z}^n$. {\it Defect } $d({\cal E},Λ)$ of the standard basis $ {\cal E}$ of ${\mathbb Z}^n$ ($n$ unit vectors going in the directions of the coordinate axes) is defined as the smallest integer $d$ such that certain $ (n-d) $ vectors from $ {\cal E} $ together with some $d$ vectors from the lattice $Λ$ form a basis of $Λ$. Let $||...||$ be $L^1$-norm on $\mathbb{Q}^n$. Suppose t…
Lattice-based cryptography relies on generating random bases which are difficult to fully reduce. Given a lattice basis (such as the private basis for a cryptosystem), all other bases are related by multiplication by matrices in $GL(n,\mathbb{Z})$. We compare the strengths of various methods to sample random elements of $GL(n,\mathbb{Z})$, finding some are stronger than others with respect to the problem of recognizing rotations of the $\mathbb{Z}^n$ lattice. In particular, the standard algorithm of multiplying unipotent generators together (as implemented in Magma's RandomSLnZ command) genera…
Let $ω=(-1+\sqrt{-3})/2$. For any lattice $P\subseteq \mathbb{Z}^n$, $\mathcal{P}=P+ωP$ is a subgroup of $\mathcal{O}_K^n$, where $\mathcal{O}_K=\mathbb{Z}[ω]\subseteq \mathbb{C}$. As $\mathbb{C}$ is naturally isomorphic to $\mathbb{R}^2$, $\mathcal{P}$ can be regarded as a lattice in $\mathbb{R}^{2n}$. Let $P$ be a multiplicative lattice (principal lattice or congruence lattice) introduced by Rosenbloom and Tsfasman. We concatenate a family of special codes with $t_{\mathfrak{P}}^\ell\cdot(P+ωP)$, where $t_{\mathfrak{P}}$ is the generator of a prime ideal $\mathfrak{P}$ of $\mathcal{O}_K$. Ap…
We show there exists a closed locally symmetric manifold $M$ modeled on $SL_n(\mathbb R)/SO(n)$, and a non-trivial homology class in degree $dim(M)-rank(M)$ represented by a totally geodesic submanifold that contains a circle factor. As a result, the comparison map $c^k:H_b^k(M,\mathbb R)\rightarrow H^k(M,\mathbb R)$ is not surjective in degree $k=dim(M)-rank(M)$. This provides a counterpart to a result of Lafont-Wang which states that $c^k$ is always surjective in degree $k\geq dim(M)-rank(M)+2$.
Motivated by a 2019 result of Faulhuber-Steinerberger, we demonstrate that the real square lattice $\mathbb{Z}^2$ exhibits the same local, extremal property as the hexagonal lattice $Λ$, where distances of lattice points from the `deep holes' of natural fundamental domains increase under perturbation. If $Δ$ is a small perturbation of $\mathbb{Z}^2$ in the space of unimodular lattices, consider $C_r$, the set of points in $A_r$ shifted to $Δ$. If $Δ$ is a perturbation of the lattice $\mathbb{Z}^2$ with respect to the Euclidean metric, then for a fixed deep hole $p$, the summed total distance o…
We construct a torsion-free arithmetic lattice in $\mathrm{PGL}_2(\mathbb{F}_2(\!(t)\!))\times\mathrm{PGL}_2(\mathbb{F}_2(\!(t)\!))$ arising from a quaternion algebra over $\mathbb{F}_2(z)$. It is the fundamental group of a square complex with universal covering $T_3\times T_3$, a product of trees with constant valency $3$, which has minimal Euler characteristic. Furthermore, our lattice gives rise to a fake quadric over $\mathbb{F}_2(\!(t)\!)$ by means of non-archimedean uniformization.
We construct Zariski-dense surface subgroups in infinitely many commensurability classes of uniform lattices of the split real Lie groups $\operatorname{SL}(n,\mathbb{R})$, $\operatorname{Sp}(2n,\mathbb{R})$, $\operatorname{SO}(k+1,k)$, and $\operatorname{G}_2$. These subgroups are images of Hitchin representations. In particular, we show that every uniform lattice of $\operatorname{Sp}(2n,\mathbb{R})$, of $\operatorname{SO}(k+1,k)$ with $k\equiv1,2[4]$ and of $\operatorname{G}_2$ contains infinitely many mapping class group orbits of Zariski-dense Hitchin representations of fixed genus. Toget…
Small diameters and generators for arithmetic lattices in $\mathrm{SL}_2(\mathbb{R})$ and certain Ramanujan graphs
本文证明了 $\mathrm{SL}_{2}(\mathbb{R})$ 中特定算术格具有小覆盖集和小生成元，并在不依赖 Ramanujan 界限的情况下给出了某些 Ramanujan 图直径的 Ramanujan 强度界限。
Two themes associated with invariant measures on the matrix groups ${\rm SL}_N(\mathbb F)$, with $\mathbb F = \mathbb R, \mathbb C$ or $\mathbb H$, and their corresponding lattices parametrised by ${\rm SL}_N(\mathbb F)/{\rm SL}_N(\mathbb O)$, $\mathbb O$ being an appropriate Euclidean ring of integers, are considered. The first is the computation of the volume of the subset of ${\rm SL}_N(\mathbb F)$ with bounded 2-norm or Frobenius norm. Key here is the decomposition of measure in terms of the singular values. The form of the volume, for large values of the bound, is relevant to asymptotic c…
扫码添加微信好友，提出您的宝贵建议 👇
💡 备注请填写：网站反馈
发表机构由 AI 从论文文本抽取，如发现错误，欢迎反馈修正建议。