Distribution of Angles in Hyperbolic Lattices
Morten S. Risager, Jimi L. Truelsen
证明了双曲格上角的一个有效的等分布结果。利用这一点推广了F的一个结果. P. Boca.
We prove an effective equidistribution result about angles in a hyperbolic lattice. We use this to generalize a result due to F. P. Boca.
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
Pair correlation of hyperbolic lattice angles
设$ω$是上半平面上的一点，$Γ$是$\mathrm{PSL}_2(\mathbb{R})$的一个离散有限余体积子群。我们猜想一个明确的公式为对相关的角度之间的测地线的格子$Γω$，与越来越大的球中心在$ω$。我们证明了这个猜想对$Γ=\mathrm{PSL}_2(\mathbb{Z})$和$ω$的一个椭圆点。
On the Pair Correlation Density for Hyperbolic Angles
设$Γ< \mathrm{PSL}_2(\mathbb{R})$是上半平面中的一个晶格，而$ω\in \mathbb{H}$是上半平面中的一个点。本文证明了以$ω$为中心的球越来越大的格点$Γω$的测地射线间的夹角集合的对关联密度函数的存在性，并给出了其显式表达式，从而证明了Boca-Popa-Zaharescu的一个猜想.
We study lattice points on hyperbolic circles centred at Heegner points of class number one. Our main result is that, on a density one subset of radii tending to infinity, the angles of such points equidistribute on the unit circle. To prove this, we establish a connection between lattice points and algebraic integers in the associated field having norm of a special form and satisfying a congruence condition. As a by-product of this, we obtain an explicit formulation of the classical hyperbolic circle problem as a shifted convolution sum for the function that counts the number of algebraic int…
Hyperbolic lattice-point counting and modular symbols
对于$\slr$的余紧群$\G$，我们固定一个实非零调和1-形式$α$。在模符号$\modsymγ{\a}$的限制下，研究了$\G$的双曲格计数问题的渐近性.我们证明了模块化符号的归一化值，当根据这个计数排序时，具有高斯分布。
For $Γ$ a cocompact or cofinite Fuchsian group, we study the hyperbolic lattice point problem in conjugacy classes, which is a modification of the classical hyperbolic lattice point problem. We use large sieve inequalities for the Riemann surfaces $Γ\backslash \mathbb H$ to obtain average results for the error term, which are conjecturally optimal. We give a new proof of the error bound $O(X^{2/3})$, due to A. Good. For $\hbox{SL}(2,{\mathbb Z})$ we interpret our results in terms of indefinite quadratic forms.
We study the fine distribution of lattice points lying on expanding circles in the hyperbolic plane $\mathbb{H}$. The angles of lattice points arising from the orbit of the modular group $PSL_{2}(\mathbb{Z})$, and lying on hyperbolic circles, are shown to be equidistributed for generic radii. However, the angles fail to equidistribute on a thin set of exceptional radii, even in the presence of growing multiplicity. Surprisingly, the distribution of angles on hyperbolic circles turns out to be related to the angular distribution of $\mathbb{Z}^2$-lattice points (with certain parity conditions) …
Zeta functions of quadratic lattices of a hyperbolic plane
本文研究了双曲平面上二次格满秩子格真等价类的Dirichlet级数，推导出相关Zeta函数的显式公式与组合计算方法，并揭示了大量真类为单格类的解析性质。
We give a short and purely bilinear proof of the fact that two chains of $p$-elementary lattices with quadratic form or alternating bilinear form over the $p$-adic integers ore more generally over a complete discrete valuation ring have common hyperbolic bases. This fact, which is useful for the study of Bruhat-Tits buildings, has been proven before with different methods by Abramenko and Nebe and by Frisch.
It is well known that the angles in a lattice acting on hyperbolic $n$-space become equidistributed. In this paper we determine a formula for the pair correlation density for angles in such hyperbolic lattices. Using this formula we determine, among other things, the asymptotic behavior of the density function in both the small and large variable limits. This extends earlier results by Boca, Pasol, Popa and Zaharescu and Kelmer and Kontorovich in dimension 2 to general dimension $n$. Our proofs use the decay of matrix coefficients together with a number of careful estimates, and lead to effect…
The hyperbolic lattice counting problem in large dimensions
本文研究大维度双曲空间中余紧格作用下的圆问题误差项平均行为，给出局部平均上界并证明均值与二阶矩的Ω结果。
On the distribution of angles between geodesic rays associated with hyperbolic lattice points
对于上半平面中的每两个点$z_0,z_1$，考虑主同余群$Γ(N)$中的所有元素$γ$，通过分数线性变换作用于上半平面，使得$z_1$和$γz_0$之间的双曲距离至多为$R>0$。本文研究了测地线$[z_1,γz_0]$之间的夹角分布为$R\to \infty$，证明了极限分布的存在与$N$无关，并明确地计算了它，当$z_1=z_0$时，它是区间$[-π/2,π/2]$上的均匀分布.
It is well known that the orbit of a lattice in hyperbolic $n$-space is uniformly distributed when projected radially onto the unit sphere. In the present work, we consider the fine-scale statistics of the projected lattice points, and express the limit distributions in terms of random hyperbolic lattices. This provides in particular a new perspective on recent results by Boca, Popa, and Zaharescu on 2-point correlations for the modular group, and by Kelmer and Kontorovich for general lattices in dimension $n=2$.
On symmetries of hyperbolic lattices of large rank
本文研究了大秩双曲晶格的对称性，证明了其例外晶格为平凡，并指出此类晶格具有最大Salem次数的对称性。
Automorphism Groups of Hyperbolic Lattices
基于J.Opgenorth引入的对偶锥的概念，给出了一个计算具有双曲双线性型格的自同构群生成系的算法.
Hyperbolic lattice point counting in unbounded rank
本文通过谱分析方法，得到了SL(n, Z)中高度不超过T的矩阵计数渐近公式，显著改进了误差项，适用于小秩和大秩情形。
We prove that the covolume of any quasi-arithmetic hyperbolic lattice (a notion that generalizes the definition of arithmetic subgroups) is a rational multiple of the covolume of an arithmetic subgroup. As a corollary, we obtain a good description for the shape of the volumes of most of the known hyperbolic n-manifolds with n>3.
A hyperbolic lattice is called \textit{$1.2$-reflective} if the subgroup of its automorphism group generated by all $1$- and $2$-reflections is of finite index. The main result of this article is a complete classification of $1.2$-reflective maximal anisotropic lattices of rank $4$.
We establish effective counting and equidistribution results for lattice points in families of domains in hyperbolic spaces, of any dimension and over any field. The domains we focus on are defined as product sets with respect to the Iwasawa decomposition. Several classical Diophantine problems can be reduced to counting lattice points in such domains, including distribution of shortest solution to the gcd equation, and angular distribution of primitive vectors in the plane. We give an explicit and effective solution to these problems, and extend them to imaginary quadratic number fields. Furt…
On hyperbolic lattices whose reduced automorphism group is virtually abelian
本文分类了对称群为虚拟阿贝尔群的整数格，进而完成虚拟阿贝尔自同构群 K3 曲面的分类，并在附录给出正交群弱逼近算法以判定不定格等距性。
A hyperbolic lattice is called \textit{$(1{,}2)$-reflective} if its automorphism group is generated by $1$- and $2$-reflections up to finite index. In this paper we prove that the fundamental polyhedron of a $\mathbb{Q}$-arithmetic cocompact reflection group in the three-dimensional Lobachevsky space contains an edge such that the distance between its framing faces is small enough. Using this fact we obtain a classification of $(1{,}2)$-reflective anisotropic hyperbolic lattices of rank $4$.
On minimal shapes and isoperimetric constants in hyperbolic lattices
本文研究了超几何晶格中最小周长形状的特性，证明了周长与面积比趋近于Häggström-Jonasson-Lyons计算的等周常数。
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。