Related papers: Lattice Point Asymptotics and Volume Growth on Tei…
A Teichmuller lattice is the orbit of a point in Teichmuller space under the action of the mapping class group. We show that the proportion of lattice points in a ball of radius r which are not pseudo-Anosov tends to zero as r tends to…
We study the asymptotic behaviour of simply connected, Riemannian manifolds $X$ of strictly negative curvature admitting a non-uniform lattice $\Gamma$. If the quotient manifold $\bar X= \Gamma \backslash X$ is asymptotically $1/4$-pinched,…
Athreya, Bufetov, Eskin and Mirzakhani have shown the number of mapping class group lattice points intersecting a closed ball of radius $R$ in Teichm\"{u}ller space is asymptotic to $e^{hR}$, where $h$ is the dimension of the…
Athreya, Bufetov, Eskin and Mirzakhani have shown the number of mapping class group lattice points intersecting a closed ball of radius $R$ in Teichm\"{u}ller space is asymptotic to $e^{hR}$, where $h$ is the dimension of the…
Counting lattice points inside a ball of large radius in Euclidean space is a classical problem in analytic number theory, dating back to Gauss. We propose a variation on this problem: studying the asymptotics of the measure of an integer…
In this note, we derive an asymptotically sharp upper bound on the number of lattice points in terms of the volume of centrally symmetric convex bodies. Our main tool is a generalization of a result of Davenport that bounds the number of…
We announce new results concerning the asymptotic behavior of the Betti numbers of higher rank locally symmetric spaces as their volumes tend to infinity. Our main theorem is a uniform version of the L\"uck Approximation Theorem…
A condition is given, under which a general lattice point counting function is asymptotic to the corresponding ball volume growth function. This is then used to give height asymptotics in the style of the Batyrev-Manin Conjecture for…
In this article, we consider a closed rank one $C^\infty$ Riemannian manifold $M$ of nonpositive curvature and its universal cover $X$. Let $b_t(x)$ be the Riemannian volume of the ball of radius $t>0$ around $x\in X$, and $h$ the…
In this paper we study the systole growth of arithmetic locally symmetric spaces up congruence covers and show that this growth is at least logarithmic in volume. This generalizes previous work of Buser and Sarnak as well as Katz, Schaps…
This paper concerns the lattice counting problem for the mapping class group of a surface $S$ acting on Teichm\"uller space with the Teichm\"uller metric. In that problem the goal is to count the number of mapping classes that send a given…
We consider random lattices taken from the general symplectic ensemble and count the number of lattice points of a typical lattice in nested families $B_t$ of certain Borel sets. Our main result is that for almost every general symplectic…
We study the asymptotic behavior of extremal length along Teichm\"uller rays. Specifically, we determine the limit of extremal length along a Teichm\"uller ray and obtain an explicit expression for this limit, which complements a related…
In this paper we prove that the Euler equation describing the motion of an ideal fluid in $\R^d$ is well-posed in a class of functions allowing spatial asymptotic expansions as $|x|\to\infty$ of any a priori given order. These asymptotic…
Let $\Gamma$ be an irreducible lattice in $\PSL_2(\RR)^d$ ($d\in\NN$) and $z$ a point in the $d$-fold direct product of the upper half plane. We study the discrete set of componentwise distances ${\bf D}(\Gm,z)\subset \RR^d$ defined in (1).…
We investigate lattice-counting problems associated with symplectic forms from the perspective of homogeneous dynamics. In the qualitative direction, we establish an analog of Margulis theorem for symplectic forms, proving density results…
The variance of the number of lattice points inside the dilated bounded set rD with random position in R^d has asymptotics r^(d-1) if the rotational quadratic average of the modulus of the Fourier transform of the set is O(r^(-d-1)). The…
A nonuniform Neumann boundary-value problem is considered for the Poisson equation in a thin $3D$ aneurysm-type domain that consists of thin curvilinear cylinders that are joined through an aneurysm of diameter $\mathcal{O}(\varepsilon).$ A…
We study the volume-distance-ratio (VDR) asymptote at the past timelike boundary point for spatially flat FLRW spacetime with scale factor $a(t) = t^{\alpha}$, and spatially hyperbolic FLRW spacetime with scale factor $a(t) = a_0…
In this note we prove that the volume of a causal diamond associated with an inertial observer in asymptotically de Sitter 4-dimensional space-time is monotonically increasing function of cosmological time. The asymptotic value of the…