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 $j\geq1$.
Categories: math.NT math.PR