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

Abstract

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[\textit{Ann. Inst. Fourier}, 2015]{GNT15}. The case of particular interest is when the gauge is taken as the Cygan--Korányi norm and the error term reads: $${\mathcal{E}}_q(t)=\#\left({\mathbb{Z}}^{2q+1}\cap {\mathcal{B}}_t\right)-\mathrm{vol}({\mathcal{B}}_1)t^{2q+2},$$ with ${\mathcal{B}}_t=\{(v,w)\in {\mathbb{H}}^q:(|v{|}^4+w^2{)}^{1/4}\le t\}$, which is closely related to the Gauss circle problem. When $q\ge 3$, Gath \cite[\textit{Ann. Sc. Norm. Super. Pisa Cl. Sci.}, 2022]{Gat22} improved upon \cite[]{GNT15} by showing that $|{\mathcal{E}}_q(t)|\lesssim t^{2q-1+1/3}$ 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 $|{\mathcal{E}}_q(t)|\lesssim t^{2q-1+241/753}$ 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.

17 pages; comments are welcome