通过Ratner定理的连续低对horocycles的等分布

Effective Equidistribution of Closed horocycles for Geometrically finite surfaces

For a complete hyperbolic surface whose fundamental group is finitely generated and has critical exponent bigger than 1/2, we obtain an effective equidistribution of closed horocycles in its unit tangent bundle. This extends a result of Sarnak in 1981 for surfaces of finite area. We also discuss its applications in Affine sieves.

Density and Equidistribution of One-Sided Horocycles of a Geometrically Finite Hyperbolic Surface

On geometrically finite negatively curved surfaces, we give necessary and sufficient conditions for a one-sided horocycle $(h^s u)_{s\ge 0}$ to be dense in the nonwandering set of the geodesic flow. We prove that all dense one-sided orbits $(h^su)_{s\ge 0}$ are equidistributed, extending results of [Bu] and [Scha2] where symmetric horocycles $(h^su)_{s\in\R}$ were considered.

Effective equidistribution of rational points on horocycle sections in $\text{ASL}(2,\mathbb{Z}) \backslash \text{ASL}(2,\mathbb{R}) $

In this paper we prove an effective equidistribution result for both primitive and non-primitive points on certain expanding horocycle sections in ASL$(2,\mathbb{Z}) \backslash $ASL$(2,\mathbb{R})$. This provides an effective version of a result recently proven by the author and generalises recent work by Einsiedler, Luethi and Shah who prove similar results in the context of the unit tangent bundle of the modular surface.

Horocycle dynamics in rank one invariant subvarieties I: weak measure classification and equidistribution

Let M be an invariant subvariety in the moduli space of translation surfaces. We contribute to the study of the dynamical properties of the horocycle flow on M. In the context of dynamics on the moduli space of translation surfaces, we introduce the notion of a 'weak classification of horocycle invariant measures' and we study its consequences. Among them, we prove genericity of orbits and related uniform equidistribution results, asymptotic equidistribution of sequences of pushed measures, and counting of saddle connection holonomies. As an example, we show that invariant varieties of rank on…

将有效Ratner等分布与斜移矩阵的半圆律联系起来

Linking effective Ratner equidistribution to the semicircle law for skew-shift matrices

本文通过有效Ratner等分布定理，证明斜移轨道定义的大Hermitian矩阵的经验谱分布以最优多项式速率收敛到Wigner半圆律，并得到矩形矩阵的Marchenko-Pastur律。

Ratner等分布定理对$\mathrm{SL}(3,\mathbb{R})$的有效版本

Effective version of Ratner's equidistribution theorem for $\mathrm{SL}(3,\mathbb{R})$

本文在自然丢番图条件下，证明了SL(3,R)/SL(3,Z)中单幂轨道的Ratner等分布定理的有效版本，给出了定量收敛结果。

Quantitative equidistribution of horocycle push-forwards of transverse arcs

Let $M = Γ\backslash \text{SL}(2,\mathbb{R})$ be a compact quotient of $\text{SL}(2,\mathbb{R})$ equipped with the normalized Haar measure $\text{vol}$, and let $\{h_t\}_{t \in \mathbb{R}}$ denote the horocycle flow on $M$. Given $p \in M$ and $W \in \mathfrak{sl}_2(\mathbb{R}) \setminus \{0\}$ not parallel to the generator of the horocycle flow, let $γ_{p}^W$ denote the probability measure uniformly distributed along the arc $s \mapsto p \exp(sW)$ for $0\leq s \leq 1$. We establish quantitative estimates for the rate of convergence of $[(h_t)_{\ast} γ_{p}^W](f)$ to $\text{vol}(f)$ for suffici…

Horocycle flow on negative variable curvature surface is standard

We provide a new proof that the horocycle flow preserving the Margulis measure on a variable negative curvature surface is standard. This was first proved by Ratner. The main purpose of this note is to provide a simplified case of the arguments for Kakutani equivalence of unipotent flows on homogeneous spaces, which have similar but more complicated structures, as well as illustrate the versatility of the method by applying it to a non-homogeneous flow.

An Effective Ratner Equidistribution Result for ASL(2,R)

Let G=ASL(2,R) be the affine special linear group of the plane, and set Gamma=ASL(2,Z). We prove a polynomially effective asymptotic equidistribution result for the orbits of a 1-dimensional, non-horospherical unipotent flow on Gamma\G.

Eisenstein series and equidistribution of Lebesgue probability measures on compact leaves of the horocycle foliations of Bianchi 3-orbifolds

Inspired by the works of Zagier, we study the probability measures $ν(t)$ with support on the flat tori which are the compact orbits of the maximal unipotent subgroup acting holomorphically on the positive orthonormal frame bundle $\mathcal{F}({M}_D)$ of 3-dimensional hyperbolic Bianchi orbifolds ${M}_D=\mathbb{H}^3/\widetildeΓ_D$, of finite volume and with only one cusp. Here $Γ_D=PSL(2, \mathcal{O})$, where $\mathcal{O}$ is the ring of integers of an imaginary quadratic field of class number one.

双曲曲面的 Abel 覆盖：谱的等分布与 horocycle 流的无限体积混合渐近性

Abelian covers of hyperbolic surfaces: equidistribution of spectra and infinite volume mixing asymptotics for horocycle flows

本文研究紧双曲曲面的 Abel 覆盖，建立 $\mathbb{Z}^d$ 覆盖上 horocycle 流相关函数的渐近展开以证明强 Krickeberg 混合，并证明有限 Abel 覆盖序列中 Casimir 算子谱测度弱收敛于绝对连续测度。

Effective equidistribution of horocycle lifts

We give a rate of equidistribution of lifts of horocycles from the space $\mathrm{SL}(2,\mathbb Z)\backslash \mathrm{SL}(2, \mathbb R)$ to the space $\mathrm{ASL}(2,\mathbb Z)\backslash \mathrm{ASL}(2,\mathbb R)$, making effective a theorem of Elkies and McMullen. This result constitutes an effective version of Ratner's measure classification theorem for measures supported on general horocycle lifts. The method used relies on Weil's resolution of the Riemann hypothesis for function fields in one variable and generalizes the approach of Strömbergsson to the case of linear lifts and that of Brow…

模曲面上沿扩张闭 horocycle（ horocycle 轨道）的有理点平移

Translates of rational points along expanding closed horocycles on the modular surface

该研究针对模曲面上沿扩张闭horocycle的有理点平移分布问题，用谱方法证明多项式扩张速率下任意平移的等分布性，同时得到稍快扩张下的非等分布结果，还通过显式极限测度给出任意快扩张时的两类结论，本原有理点也有类似结果。

Shrinking target equidistribution of horocycles in cusps

Consider a shrinking neighborhood of a cusp of the unit tangent bundle of a noncompact hyperbolic surface of finite area, and let the neighborhood shrink into the cusp at a rate of $T^{-1}$ as $T \rightarrow \infty$. We show that a closed horocycle whose length $\ell$ goes to infinity or even a segment of that horocycle becomes equidistributed on the shrinking neighborhood when normalized by the rate $T^{-1}$ provided that $T/\ell \rightarrow 0$ and, for any $δ>0$, the segment remains larger than $\max\left\{T^{-1/6},\left(T/\ell\right)^{1/2}\right\}\left(T/\ell\right)^{-δ}$. We also have an e…

Uniform bounds for period integrals and sparse equidistribution

Let $M=Γ\backslash\mathrm{PSL}(2,\mathbb{R})$ be a compact manifold, and let $f\in C^\infty(M)$ be a function of zero average. We use spectral methods to get uniform (i.e. independent of spectral gap) bounds for twisted averages of $f$ along long horocycle orbit segments. We apply this to obtain an equidistribution result for sparse subsets of horocycles on $M$.

Effective Ratner theorem for ASL(2,R) and gaps in \sqrt{n} modulo 1

Let G=ASL(2,R) be the affine special linear group of the plane, and set Gamma=ASL(2,Z). Building on recent work of Strömbergsson we prove a rate of equidistribution for the orbits of a certain 1-dimensional unipotent flow of Gamma\G, which projects to a closed horocycle in the unit tangent bundle to the modular surface. We use this to answer a question of Elkies and McMullen by making effective the convergence of the gap distribution of \sqrt{n} modulo 1.

Equidistribution of Horocyclic Flows on Complete Hyperbolic Surfaces of Finite Area

We provide a self-contained, accessible introduction to Ratner's Equidistribution Theorem in the special case of horocyclic flow on a complete hyperbolic surface of finite area. This equidistribution result was first obtained in the early 1980s by Dani and Smillie and later reappeared as an illustrative special case of Ratner's work on the equidistribution of unipotent flows in homogeneous spaces. We also prove an interesting probabilistic result due to Breuillard: on the modular surface an arbitrary uncentered random walk on the horocycle through almost any point will fail to equidistribute, …

Primitive rational points on expanding horocycles in products of the modular surface with the torus

We prove effective equidistribution of primitive rational points and of primitive rational points defined by monomials along long horocycle orbits in products of the torus and the modular surface. This answers a question posed in joint work by the first and the last named author with Shahar Mozes and Uri Shapira. Under certain congruence conditions we prove the joint equidistribution of conjugate rational points in the two-torus and the modular surface.

Distribution modulo one and Ratner's theorem

Measure rigidity is a branch of ergodic theory that has recently contributed to the solution of some fundamental problems in number theory and mathematical physics. Examples are proofs of quantitative versions of the Oppenheim conjecture, related questions on the spacings between the values of quadratic forms, a proof of quantum unique ergodicity for certain classes of hyperbolic surfaces, and an approach to the Littlewood conjecture on the nonexistence of multiplicatively badly approximable numbers. In these lectures we discuss a few simple applications of one of the central results in measur…

关于$PSL_2(\mathbb{R})$的商空间中多项式序列的等分布

On equidistribution of polynomial sequences in quotients of $PSL_2(\mathbb{R})$

该研究证明$PSL_2(\mathbb{R})$中任意格对应的商空间上，带指数$1+\gamma$（$\gamma$小于某正常数）的horocycle流多项式序列等分布，推广了余紧格情形的已有结果，推进了Shah猜想的研究。

2-幂零波兰群中的等分布与三重受限和集

Equidistribution in 2-Nilpotent Polish Groups and triple restricted sumsets

本文研究2-幂零局部紧波兰群的等分布定理，证明了在任意可数离散阿贝尔群中，正密度子集存在三重受限和集。

Joint equidistribution on the product of the circle and the unit tangent bundle of the modular surface

We use spectral method to prove a joint equidistribution of primitive rational points and the same along expanding horocycle orbits in the products of the circle and the unit cotangent bundle of the modular surface. This result explicates the error bound in a recent work of Einsiedler, Luethi, and Shah \cite[Theorem $1.1$]{ELS}. The error is sharp upon the best known progress towards the Ramanujan conjecture at the finite places for the modular surface.

多项式有效等分布对于单变量上三角子群在SL₂因子积中的等分布

Polynomially effective equidistribution for unipotent orbits in products of $\mathrm{SL}_2$ factors

本文提出了一种多项式速率的有效等分布定理，针对单变量上三角子群在SL₂因子积中的等分布问题，通过引入中间群和障碍物分析来克服等分布的挑战。