Related papers: The hyperbolic lattice counting problem in large d…

A classic theorem of Kazhdan and Margulis states that for any semisimple Lie group without compact factors, there is a positive lower bound on the covolume of lattices. H. C. Wang's subsequent quantitative analysis showed that the…

We show that an isometric action of a torsion-free uniform lattice $\Gamma$ on hyperbolic space $\mathbb{H}^n$ can be metrically approximated by geometric actions of $\Gamma$ on $\mathrm{CAT}(0)$ cube complexes, provided that either $n$ is…

Using, and extending, striking inequalities by V.V. Ivanov on the down-crossings of monotone functions and ergodic sums, we give universal bounds on the probability of finding oscillations of observables in 1-dimensional lattice gases in…

Let $E(x;\omega)$ be the error term for the number of integer lattice points lying inside a $3$-dimensional Cygan-Kor\'anyi spherical shell of inner radius $x$ and gap width $\omega(x)>0$. Assuming that $\omega(x)\to0$ as $x\to\infty$, and…

We prove a sharp bound for the remainder term of the number of lattice points inside a ball, when averaging over a compact set of (not necessarily unimodular) lattices, in dimensions two and three. We also prove that such a bound cannot…

For $\Gamma$ a cofinite Fuchsian group, and $l$ a fixed closed geodesic, we study the asymptotics of the number of those images of $l$ that have a prescribed orientation and distance from $l$ less than or equal to $X$. Using a new relative…

We evaluate the variance of the number of lattice points in a small randomly rotated spherical ball on a surface of 3-dimensional sphere centered at the origin. Previously, Bourgain, Rudnick, and Sarnak showed conditionally on the…

Heinz Huber (1956) considered the following problem on the the hyperbolic plane H. Consider a strictly hyperbolic subgroup of automorphisms on H with compact quotient, and choose a conjugacy class in this group. Count the number of vertices…

We use classical methods from analytic number theory to resolve the lattice point counting problem on the first Heisenberg group, in the case where the gauge function is taken to be the Cygan-Kor$\acute{a}$nyi Heisenberg-norm…

We use spectral analysis to give an asymptotic formula for the number of matrices in SL(n, Z) of height at most T with strong error terms, far beyond the previous known, both for small and large rank.

For quotients of the $n+1$-dimensional hyperbolic space by a convex co-compact group $\Gamma$, we obtain a formula relating the renormalized trace of the wave operator with the resonances of the Laplacian and some conformal invariants of…

We establish an error estimate for counting lattice points in Euclidean norm balls (associated to an arbitrary irreducible linear representation) for lattices in simple Lie groups of real rank at least two. Our approach utilizes refined…

We study lattice embeddings for the class of countable groups $\Gamma$ defined by the property that the largest amenable uniformly recurrent subgroup $A_\Gamma$ is continuous. When $A_\Gamma$ comes from an extremely proximal action and the…

For $\Gamma$ a relatively hyperbolic group, we construct a model for the universal space among $\Gamma$-spaces with isotropy on the family VC of virtually cyclic subgroups of $\Gamma$. We provide a recipe for identifying the maximal…

We establish central limit theorems for an action of a group G on a hyperbolic space X with respect to the counting measure on a Cayley graph of G. Our techniques allow us to remove the usual assumptions of properness and smoothness of the…

We prove limit theorems for the greatest common divisor and the least common multiple of random integers. While the case of integers uniformly distributed on a hypercube with growing size is classical, we look at the uniform distribution on…

We consider equidistribution of angles for certain hyperbolic lattice points in the upper half-plane. Extending work of Friedlander and Iwaniec we show that for the full modular group equidistribution persists for matrices with…

We consider scattering by star-shaped obstacles in hyperbolic space and show that resonances satisfy a universal bound $\mathrm{Im}\,\lambda \leq -\frac{1}{2}$ which is optimal in dimension $2$. In odd dimensions we also show that…

We consider a class of Hamiltonians in $L^2(\R^2)$ with attractive interaction supported by piecewise $C^2$ smooth loops $\Gamma$ of a fixed length $L$, formally given by $-\Delta-\alpha\delta(x-\Gamma)$ with $\alpha>0$. It is shown that…

Let $\Gamma$ 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 $\Gamma\backslash G$. Our method of proof relies…