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

Abstract

We consider the problem of estimating the error term ${\mathcal{E}}_q(x)=|{\mathbb{Z}}^{2q+1}\cap {\delta }_x\mathcal{B}|-\text{vol}\left(\mathcal{B}\right)x^{2q+2}$ which occurs in the counting of lattice points in Heisenberg dilates of the Cygan-Kor{á}nyi ball. We prove three type of results regarding the order of magnitude of ${\mathcal{E}}_q(x)$, which are valid for any $q\ge 3$. 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 ${\mathcal{E}}_q(x)=\Omega \left(x^{2q-1}\right(\log x{)}^{1/4}\left(\log \log x{)}^{1/8}\right)$. Consequently, we obtain the lower bound ${\kappa }_q=\sup \{\alpha >0:|{\mathcal{E}}_q(x)|\ll x^{2q+2-\alpha }\}\ge \frac{8}{3}$ for $q\ge 3$, and conjecture that ${\kappa }_q=3$

49 pages. Comments are welcome

References in corpus (1)

• On the best possible exponent for the error term in the lattice point counting problem on the first Heisenberg group