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

Abstract

Abstract We investigate the fluctuations in the number of integral lattice points on the Heisenberg groups which lie inside a Cygan–Korányi ball of large radius. Let $$ \mathcal{E}_{q}(x)=\big|\mathbb{Z}^{2q+1}\cap\delta_{x}\mathcal{B}\big|-\textit{vol}\big(\mathcal{B}\big)x^{2q+2}\,, $$ be the error term which occurs for this lattice point counting problem on the $(2q + 1)$-dimensional Heisenberg group $\mathbb{H}_{q}$, where $\mathcal{B}=\{\mathbf{u}\in\mathbb{R}^{2q+1}:|\mathbf{u}|_{{Cyg}}\leq\,1\}$ is the unit ball with respect to the Cygan–Korányi norm $$ |\mathbf{u}|_{{Cyg}}=\Big(\Big(u^{2}_{1}+\cdots+u^{2}_{2q}\Big)^{2}+u^{2}_{2q+1}\Big)^{1/4}\,, $$ and $\big\{\delta_{x}\big\}_{x\gt0}$ are the Heisenberg dilations given by $\delta_{x}\mathbf{u}=(xu_{{1}},\ldots, xu_{{2q}},x^{2}u_{{2q+1}})$. For $q\geq\,3$, we prove that the suitably normalized error term $\mathcal{E}_{q}(x)/x^{2q-1}$ has a limiting value distribution in the sense that there exists a probability density $\mathcal{P}_{q}(\alpha)$ such that for any interval $\mathcal{I}$, $$ \lim_{X\to\infty}\frac{1}{X}\textit{meas}\Big\{x\in[X,2X]:\mathcal{E}_{q}(x)/x^{2q-1}\in\mathcal{I}\Big\}=\int\limits_{\mathcal{I}}\mathcal{P}_{q}(\alpha)\textit{d}\alpha\,. $$ The density $\mathcal{P}_{q}(\alpha)$ is an entire function of α which satisfies, for any non-negative integer $j\geq\,0$ and any real α, $|\alpha|$ sufficiently large in terms of j and q, the upper bound $$ \begin{split} \big|\mathcal{P}^{(j)}_{q}(\alpha)\big|\leq\exp{\Big(-|\alpha|^{4-\beta/\log\log{|\alpha|}}\Big)}\,, \end{split} $$ where β > 0 is an absolute constant. In addition, we prove that $\int_{-\infty}^{\infty}\alpha\mathcal{P}_{q}(\alpha)\textit{d}\alpha=0$ and $\int_{-\infty}^{\infty}\alpha^{3}\mathcal{P}_{q}(\alpha)\textit{d}\alpha\lt0$, and we derive an explicit formula for the j-th integral moment of the density $\mathcal{P}_{q}(\alpha)$ for any integer $j\geq\,1$.