D. Y. Kleinbock, G. A. Margulis
We prove an effective version of a result obtained recently by Kleinbock and Weiss on equidistribution of expanding translates of orbits of horospherical subgroups in the space of lattices.
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Let $G$ be a semisimple Lie group and $Γ$ a lattice in $G$. We generalize a method of Burger to prove precise effective equidistribution results for translates of pieces of horospheres in the homogeneous space $Γ\backslash G$.
Let $Γ$ be a lattice in $G=\mathrm{SL}(2,\mathbb{C})$. We give an effective equidistribution result with precise error terms for expanding translates of pieces of horospherical orbits in $Γ\backslash G$. Our method of proof relies on the theory of unitary representations.
Given a homogeneous space $X = G/Γ$ with $G$ containing the group $H = (\mathrm{SO}(n,1))^k$. Let $x\in X$ such that $Hx$ is dense in $X$. Given an analytic curve $ϕ: I=[a,b] \rightarrow H$, we will show that if $ϕ$ satisfies certain geometric condition, then for a typical diagonal subgroup $A =\{a(t): t \in \mathbb{R}\} \subset H$ the translates $\{a(t)ϕ(I)x: t >0\}$ of the curve $ϕ(I)x$ will tend to be equidistributed in $X$ as $t \rightarrow +\infty$. The proof is based on the study of linear representations of $\mathrm{SO}(n,1)$ and $H$.
Let $(X,\mathfrak{B},μ)$ be a Borel probability space. Let $T_n: X\rightarrow X$ be a sequence of continuous transformations on $X$. Let $ν$ be a probability measure on $X$ such that $\frac{1}{N}\sum_{n=1}^N (T_n)_\ast ν\rightarrow μ$ in the weak-$\ast$ topology. Under general conditions, we show that for $ν$ almost every $x\in X$, the measures $\frac{1}{N}\sum_{n=1}^N δ_{T_n x}$ get equidistributed towards $μ$ if $N$ is restricted to a set of full upper density. We present applications of these results to translates of closed orbits of Lie groups on homogeneous spaces. As a corollary, we prov…
Effective equidistribution of expanding translates in the space of affine lattices
本文证明了任意$d\ge 2$维仿射格空间中膨胀平移的多项式有效等分布结果。
Einsiedler, Mozes, Shah and Shapira [Compos. Math. 152 (2016), 667-692] prove an equidistribution theorem for rational points on expanding horospheres in the space of d-dimensional Euclidean lattices, with d>2. Their proof exploits measure classification results, but provides no insight into the rate of convergence. We pursue here an alternative approach, based on harmonic analysis and Weil's bound for Kloosterman sums, which in dimension d=3 yields an effective estimate on the rate of convergence.
We study equidistribution of certain subsets of periodic orbits for subshifts of finite type. Our results solely rely on the growth of these subsets. As a consequence, effective equidistribution results are obtained for both hyperbolic diffeomorphisms and expanding maps on compact manifolds.
Equidistribution of primitive rational points on expanding horospheres
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Equidistribution of expanding translates of smooth curves in homogeneous spaces under the action of a product of SO(n,1)'s
本文证明了在多个SO(n,1)乘积群作用下，光滑曲线的扩张平移在避开可数代数障碍后于轨道闭包中等分布，方法结合Ratner定理、Kempf几何不变量理论与线性化技术。
We consider improvements of Dirichlet's Theorem on space of matrices $M_{m,n}(R)$. It is shown that for a certain class of fractals $K\subset [0,1]^{mn}\subset M_{m,n}(R)$ of local maximal dimension Dirichlet's Theorem cannot be improved almost everywhere. This is shown using entropy and dynamics on homogeneous spaces of Lie groups.
We study the general problem of equidistribution of expanding translates of an analytic curve by an algebraic diagonal flow on the homogeneous space $G/Γ$ of a semisimple algebraic group $G$. We define two families of algebraic subvarieties of the associated partial flag variety $G/P$, which give the obstructions to non-divergence and equidistribution. We apply this to prove that for Lebesgue almost every point on an analytic curve in the space of $m\times n$ real matrices whose image is not contained in any subvariety coming from these two families, the Dirichlet's theorem on simultaneous Dio…
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Let $H = \mathrm{SO}(n,1)$ and $A =\{a(t) : t \in \mathbb{R}\}$ be a maximal $\mathbb{R}$-split Cartan subgroup of $H$. Let $G$ be a Lie group containing $H$ and $Γ$ be a lattice of $G$. Let $x = gΓ\in G/Γ$ be a point of $G/Γ$ such that its $H$-orbit $Hx$ is dense in $G/Γ$. Let $ϕ: I= [a,b] \rightarrow H$ be an analytic curve, then $ϕ(I)x$ gives an analytic curve in $G/Γ$. In this article, we will prove the following result: if $ϕ(I)$ satisfies some explicit geometric condition, then $a(t)ϕ(I)x$ tends to be equidistributed in $G/Γ$ as $t \rightarrow \infty$. It answers the first question asked…
We prove an effective version of a result due to Einsiedler, Mozes, Shah and Shapira who established the equidistribution of primitive rational points on expanding horospheres in the space of unimodular lattices in at least $3$ dimensions. Their proof uses techniques from homogeneous dynamics and relies in particular on measure-classification theorems -- an approach which does not lend itself to effective bounds. We implement a strategy based on spectral theory, Fourier analysis and Weil's bound for Kloosterman sums in order to quantify the rate of equidistribution for a specific horospherical…
Equidistribution of expanding degenerate manifolds in the space of lattices
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In this paper we study spherical equidistribution on the space of (translates of) adelic lattices, which we apply to understand the fine-scale statistics of the directions in the set of shifted primitive lattice points. We also apply our results to the distribution of the free path lengths in the Boltzmann--Grad limit for point sets such as (possibly non-rational) translates of the lattice points all of whose coordinates are squarefree. Besides the equidistribution results for translates of expanding horospheres, a key ingredient is a probabilistic argument which allows us to tackle the techni…
Let $U$ be a horospherical subgroup of a noncompact simple Lie group $H$ and let $A$ be a maximal split torus in the normalizer of $U$. We define the expanding cone $A_U^+$ in $A$ with respect to $U$ and show that it can be explicitly calculated. We prove several dynamical results for translations of $U$-slices by elements of $A_U^+$ on finite volume homogeneous space $G/Γ$ where $G$ is a Lie group containing $H$. More precisely, we prove quantitative nonescape of mass and equidistribution of a $U$-slice. If $H$ is a normal subgroup of $G$ and the $H$ action on $G/Γ$ has a spectral gap, we pro…
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。
Margulis' Inequality for translates of horospherical orbits and applications to equidistribution
该研究借助 Eskin-Margulis-Mozes 与 Eskin-Margulis 的 Margulis 不等式，证明水平球面轨道平移的定量非发散定理，并推出数域上非真有理抛物子群内格轨道的有效等分布。
We use a dictionary between lattice point counting inside dilated d-dimensional ellipsoids (Euclidean counting) and counting of lifts of a closed horosphere that intersect a ball of increasing radius, to obtain two types of results. Firstly, via an $L^2$-integral error estimate for Euclidean counting, we prove effective equidistribution results for a family of expanding horospheres in the locally symmetric space SO(d) \ SL(d,R) / SL(d,Z). Secondly, we derive from uniform error estimates in Euclidean counting, error terms for counting SL(d,Z)-orbit points in a certain increasing family of subse…
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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.
In this article, we study an analytic curve $φ: I=[a,b]\rightarrow \mathrm{M}(n\times n, \mathbb{R})$ in the space of $n$ by $n$ real matrices, and show that if $φ$ satisfies certain geometric conditions, then for almost every point on the curve, the Diophantine approximation given by Dirichlet's Theorem is not improvable. To do this, we embed the curve into some homogeneous space $G/Γ$, and prove that under the action of some expanding diagonal flow $A= \{a(t): t \in \mathbb{R}\}$, the expanding curves tend to be equidistributed in $G/Γ$, as $t \rightarrow +\infty$. This solves a special case…
Let G be a semisimple linear algebraic group defined over rational numbers, K be a maximal compact subgroup of its real points and Γ be an arithmetic lattice. One can associate a probability measure μ(H) on Γ\G for each subgroup H of G defined over Q with no non-trivial rational characters. As G acts on Γ\G from the right, we can push-forward this measure by elements from G. By pushing down these measures to Γ\G/K, we call them homogeneous. It is a natural question to ask what are the possible weak-* limits of homogeneous measures. In the non-divergent case this has been answered by Eskin--Moz…
Recently Mohammadi and Salehi-Golsefidy gave necessary and sufficient conditions under which certain translates of homogeneous measures converge, and they determined the limiting measures in the cases of convergence. The class of measures they considered includes the maximal horospherical measures. In this paper we prove the corresponding effective equidistribution results in the space of unimodular lattices. We also prove the corresponding results for probability measures with absolutely continuous densities in rank two and three. Then we address the problem of determining the error terms in …
Effective equidistribution of primitive rational points on expanding horospheres
本文利用矩阵 Kloosterman 和界、Hecke 点有效等分布及 Rogers 积分公式，证明扩张闭 horosphere 上本原有理点的有效等分布，并给出线性同余小解与有限域非方阵计数应用。
Let $T \colon M \to M$ be a nonuniformly expanding dynamical system, such as logistic or intermittent map. Let $v \colon M \to \mathbb{R}^d$ be an observable and $v_n = \sum_{k=0}^{n-1} v \circ T^k$ denote the Birkhoff sums. Given a probability measure $μ$ on $M$, we consider $v_n$ as a discrete time random process on the probability space $(M, μ)$. In smooth ergodic theory there are various natural choices of $μ$, such as the Lebesgue measure, or the absolutely continuous $T$-invariant measure. They give rise to different random processes. We investigate relation between such processes. We sh…
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Let $G=SL_2(\mathbb R)^d$ and $Γ=Γ_0^d$ with $Γ_0$ a lattice in $SL_2(\mathbb R)$. Let $S$ be any "curved" submanifold of small codimension of a maximal horospherical subgroup of $G$ relative to an $\mathbb R$-diagonalizable element $a$ in the diagonal of $G$. Then for $S$ compact our result can be described by saying that $a^n \text{vol}_S$ converges in an effective way to the volume measure of $G/Γ$ when $n\to \infty$, with $\text{vol}_S$ the volume measure on $S$.
Let $X=\text{SL}_3(\mathbb{R})/\text{SL}_3(\mathbb{Z})$, and $g_t=\text{diag}(e^{2t}, e^{-t}, e^{-t})$. Let $ν$ denote the push-forward of the normalized Lebesgue measure on a segment of a straight line in the expanding horosphere of $\{g_t\}_{t>0}$, under the map $h\mapsto h\text{SL}_3(\mathbb{Z})$ from $\text{SL}_3(\mathbb{R})$ to $X$. We give explicit necessary and sufficient Diophantine conditions on the line for equidistribution of each of the following families of measures on $X$: (1) $g_t$-translates of $ν$ as $t\to\infty$. (2) averages of $g_t$-translates of $ν$ over $t\in[0,T]$ as $T\…
This paper introduces an abstract spectral approach to prove effective equidistribution of expanding horospheres in hyperbolic manifolds. The method, which is motivated by the approach to counting developed by (Lax-Phillips 1982), produces highly effective, explicit error terms. To exhibit the flexibility of this method we prove effective horospherical equidistribution theorems in $T^1(\mathbb{H}^{n+1})$ and in the higher rank setting, $\operatorname{SL}_n(\mathbb{R})/\operatorname{SO}_n(\mathbb{R})$.
We prove an avoidance principle for expanding translates of unipotent orbits for some semisimple homogeneous spaces. In addition, we prove a quantitative isolation result of closed orbits and give an upper bound on the number of closed orbits of bounded volume. The proofs of our results rely on the construction of a Margulis function and the theory of finite dimensional representations of semisimple Lie groups.
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该研究针对实线上具有共同压缩比的自相似测度乘积，证明了加权Khintchine-Schmidt定理的分形类似结果，通过提出锥提升方法，得到Khalil等人定理的部分有效形式。
We prove that certain sequences of periodic orbits of the diagonal group in the space of lattices equidistribute. As an application we obtain new information regarding the sequence of best approximations to certain vectors with algebraic coordinates. In order to prove these results we generalize the seminal work of Eskin Mozes and Shah about the equidistribution of translates of periodic measures from the real case to the S-arithmetic case.
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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.
Equidistribution of non-uniformly stretching translates of shrinking smooth curves and weighted Dirichlet approximation
本文研究非均匀伸缩作用下非退化光滑曲线平移在幺模格空间中的等分布，证明固定大小及收缩曲线段均趋于等分布，并由此推出加权Dirichlet逼近定理对几乎所有点不可改进，推广了Shah的解析曲线结果。
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.
In this paper, we study an analytic curve $φ: I=[a,b]\rightarrow \mathrm{M}(m\times n, \mathbb{R})$ in the space of $m$ by $n$ real matrices, and show that if $φ$ satisfies certain geometric condition, then for almost every point on the curve, the Diophantine approximation given by Dirichlet's Theorem can not be improved. To do this, we embed the curve into some homogeneous space $G/Γ$, and prove that under the action of some expanding diagonal subgroup $A= \{a(t): t \in \mathbb{R}\}$, the translates of the curve tend to be equidistributed in $G/Γ$, as $t \rightarrow +\infty$.
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Let G be the identity component of SO(n,1), acting linearly on a finite dimensional real vector space V. Consider a vector w_0 in V such that the stabilizer of w_0 is a symmetric subgroup of G or the stabilizer of the line Rw_0 is a parabolic subgroup of G. For any non-elementary discrete subgroup Gamma of G with w_0Gamma discrete, we compute an asymptotic formula for the number of points in w_0Gamma of norm at most T, provided that the Bowen-Margulis-Sullivan measure on the associated hyperbolic manifold and the Gamma skinning size of w_0 are finite. The main ergodic ingredient in our approac…
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In this paper, we show that each expanding Thurston map $f : S^2\rightarrow S^2$ has $1+ deg f$ fixed points, counted with appropriate weight, where $ deg f$ denotes the topological degree of the map $f$. We then prove the equidistribution of preimages and of (pre)periodic points with respect to the unique measure of maximal entropy $μ_f$ for $f$. We also show that $(S^2,f,μ_f)$ is a factor of the left shift on the set of one-sided infinite sequences with its measure of maximal entropy, in the category of measure-preserving dynamical systems. Finally, we prove that $μ_f$ is almost surely the w…
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The Boltzmann-Grad Limit of the Lorentz Gas in a Union of Lattices
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