On the Distribution of the Number of Lattice Points in Norm Balls on the Heisenberg Groups

TL;DR

This paper studies the distribution of lattice point counts within large Cygan-Korányi norm balls on Heisenberg groups, revealing a limiting distribution with specific smoothness and decay properties for dimensions q≥3.

Contribution

It establishes the existence and properties of a limiting distribution for normalized lattice point errors on Heisenberg groups, extending the density to entire functions and providing explicit moments.

Findings

The normalized error term converges to an absolutely continuous distribution.

The density function extends to an entire function on the complex plane.

Explicit formulas for moments of the distribution are derived.

Abstract

We investigate the fluctuations in the number of integral lattice points on the Heisenberg groups which lie inside a Cygan-Kor{\'a}nyi norm ball of large radius. Let $\mathcal{E}_{q}(x)=\big|\mathbb{Z}^{2q+1}\cap\delta_{x}\mathcal{B}\big|-\textit{vol}\big(\mathcal{B}\big)x^{2q+2}$ denote the error term which occurs for this lattice point counting problem on the Heisenberg group $\mathbb{H}_{q}$, where $\mathcal{B}$ is the unit ball in the Cygan-Kor{\'a}nyi norm and $\delta_{x}$ is the Heisenberg-dilation by $x>0$. For $q\geq3$ we consider the suitably normalized error term $\mathcal{E}_{q}(x)/x^{2q-1}$, and prove it has a limiting value distribution which is absolutely continuous with respect to the Lebesgue measure. We show that the defining density for this distribution, denoted by $\mathcal{P}_{q}(\alpha)$, can be extended to the whole complex plane $\mathbb{C}$ as an entire function of $\alpha$ and satisfies for any non-negative integer $j\geq0$ and any $\alpha\in\mathbb{R}$, $|\alpha|>\alpha_{q,j}$, the bound: \begin{equation*} \begin{split} \big|\mathcal{P}^{(j)}_{q}(\alpha)\big|\leq\exp{\Big(-|\alpha|^{4-\beta/\log\log{|\alpha|}}\Big)} {split} {equation*} where $\beta>0$ is an absolute constant. In addition, we give an explicit formula for the $j$-th integral moment of the density $\mathcal{P}_{q}(\alpha)$ for any integer $j\geq1$.