Lattice point counting in Cygan--Korányi balls on Heisenberg groups

Lattice point counting in gauge balls on the Heisenberg group ${\mathbb{H}}^q$ is a non-commutative analogue of the Euclidean multidimensional sphere problem, initiated by Garg, Nevo and Taylor \cite[$\mathit{\text{Ann.}}\mathit{\text{Inst.}}\mathit{\text{Fourier}}$, 2015]{GNT15}. The case of particular interest is when the gauge is taken as the Cygan–Korányi norm and the error term reads: $${\mathcal{E}}_q(t)=\mathrm{\#}({\mathbb{Z}}^{2q+1}\cap {\mathcal{B}}_t)-\mathrm{vol}({\mathcal{B}}_1)t^{2q+2},$$ with ${\mathcal{B}}_t=\{(v,w)\in {\mathbb{H}}^q:(\mathrm{\mid }v{\mathrm{\mid }}^4+w^2{)}^{1\mathrm{/}4}\le t\}$, which is closely related to the Gauss circle problem. When $q\ge 3$, Gath \cite[$\mathit{\text{Ann.}}\mathit{\text{Sc.}}\mathit{\text{Norm.}}\mathit{\text{Super.}}\mathit{\text{Pisa}}\mathit{\text{Cl.}}\mathit{\text{Sci.}}$, 2022]{Gat22} improved upon \cite[]{GNT15} by showing that $\mathrm{\mid }{\mathcal{E}}_q(t)\mathrm{\mid }\lesssim t^{2q-1+1\mathrm{/}3}$ and proposed the conjecture that the optimal order should be $2q-1$. In this paper, through Landau’s formula and the $5,6$-th Derivative Tests of van der Corput, we arrive at that $\mathrm{\mid }{\mathcal{E}}_q(t)\mathrm{\mid }\lesssim t^{2q-1+241\mathrm{/}753}$ for any $q\ge 4$, and recover the bound of Gath for $q=3$ up to a logarithmic factor. This, via a simpler method, provides the first progress towards Gath’s conjecture.