Lattice point counting problems on step-two nilpotent Lie groups

Abstract

We develop the theory of lattice point counting on connected and simply connected nilpotent Lie groups of step-two, endowed with the parabolic type dilation and a family of homogeneous norms $ \mathcal{N}_{\alpha,M}(x, t)=\left(|M_1x|^\alpha + |M_2t|^{\alpha / 2}\right)^{1 / \alpha}$ adapted to the dilation structure, where $\alpha>0$ and $M_1,M_2$ are invertible matrices. With appropriate notions of lattices, the domains to be counted are balls associated to these norms, and explicit counting discrepancy estimates are deduced for all possible dimensions and all $\alpha>0$. The bounds are sharp when the group center is unidimensional and $\alpha=2$, in certain rational sense. Our study also generalizes and even quantitatively improves previous results on Heisenberg groups obtained by Garg--Nevo--Taylor \cite[\textit{Ann. Inst. Fourier}, 2015]{GNT15}: (i) In dimension $5$, the exponent of logarithmic factor is lowered from $2/3$ to ${1}/{3}$ if $\alpha \in(3,4] $ or $\alpha=1$; and the factor $\log ^{2/3} R $ is dropped if $\alpha\in(2,3]$. (ii) In dimension $3$, the estimation is upgraded from $O_\epsilon(R^{ 5/2+\epsilon})$ to $O(R^{2}\log^{ 1/2} R)$ for $\alpha=1$, and to $O(R^{{19}/{8}})$ for $\alpha\in (1,2)$; and the factor $\log R$ is removed for $\alpha>4$. Moreover, as a byproduct, we extend the lattice counting near Heisenberg spheres, recently considered by Campolongo--Taylor \cite[\textit{Matematica}, 2023]{CT23} and Srivastava--Taylor \cite[\textit{J. Fourier Anal. Appl.}, 2026]{ST26}, to the above step-two group setting with arbitrary dimensional group center, where some quantitative improvements are also attained. Our method relies upon Poisson's summation formulas, oscillatory integral estimates and asymptotic properties as well as recursion formulas of Bessel functions.