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

Yoav A. Gath

Published 2020-10-02Version 1

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 $j\geq1$.