The lattice point counting problem on the Heisenberg groups

Abstract

We consider the radial and Heisenberg-homogeneous norms on the Heisenberg groups given by $N_{\alpha ,A}((z,t))={\left(|z{|}^{\alpha }+A|t{|}^{\alpha /2}\right)}^{1/\alpha }$, for $\alpha \ge 2$ and $A>0$. This natural family includes the canonical Cygan-Korányi norm, corresponding to $\alpha =4$. We study the lattice points counting problem on the Heisenberg groups, namely establish an error estimate for the number of points that the lattice of integral points has in a ball of large radius $R$. The exponent we establish for the error in the case $\alpha =2$ is the best possible, in all dimensions.