Lattice points for products of upper half planes

TL;DR

This paper investigates the distribution of lattice points in products of upper half planes, providing new asymptotic counting results in expanding regions and generalizing classical error bounds such as Selberg's.

Contribution

It introduces novel asymptotic formulas for counting lattice points in expanding hypercubes and strips, extending previous error estimates and analyzing dependence on the base point.

Findings

Asymptotic formulas for lattice point counts in expanding hypercubes.

New results on counting in expanding strips in multiple directions.

Generalization of Selberg's error term for higher dimensions.

Abstract

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 prove asymptotic results on the number of $\gm\in\Gm$ such that $d(z,\gamma z$ is contained in strips expanding in some directions and also in expanding hypercubes. The results on the counting in expanding strips are new. The results on expanding hypercubes % improve the error terms improve the existing error terms (by Gorodnick and Nevo) and generalize the Selberg error term for $d=1$. We give an asymptotic formula for the number of lattice points $\gamma z$ such that the hyperbolic distance in each of the factors satisfies $d((\gamma z)_j, z_j)\le T$. The error term, as $T \to \infty$ generalizes the error term given by Selberg for $d=1$, also we describe how the counting function depends on $z$. We also prove asymptotic results when the distance satisfies $A_j \le d((\gamma z)_j, z_j) < B_j$, with fixed $A_j < B_j$ in some factors, while in the remaining factors $0 \le d((\gamma z)_j, z_j) \le T$ is satisfied.