On an Analogue Of the Gauss Circle Problem For the Heisenberg Groups

We consider the problem of estimating the error term $\mathcal{E}_{q}(x)=\big|\mathbb{Z}^{2q+1}\cap\delta_{x}\mathcal{B}\big|-\textit{vol}\big(\mathcal{B}\big)x^{2q+2}$ which occurs in the counting of lattice points in Heisenberg dilates of the Cygan-Kor{\'a}nyi ball. We prove three type of results regarding the order of magnitude of $\mathcal{E}_{q}(x)$, which are valid for any $q\geq3$. An upper bound estimate of the form $|\mathcal{E}_{q}(x)|\ll x^{2q-2/3}$ ; A sharp second moment estimate, which shows that $\mathcal{E}_{q}(x)$ has order of magnitude $x^{2q-1}$ in mean-square ; And an $\Omega$-estimate of the form $\mathcal{E}_{q}(x)=\Omega\big(x^{2q-1}\big(\log{x}\big)^{1/4}\big(\log{\log{x}}\big)^{1/8}\big)$. Consequently, we obtain the lower bound $\kappa_{q}=\sup\big\{\alpha>0:\big|\mathcal{E}_{q}(x)\big|\ll x^{2q+2-\alpha}\big\}\geq\frac{8}{3}$ for $q\geq3$, and conjecture that $\kappa_{q}=3$