The lattice point counting problem on the Heisenberg groups

We consider the radial and Heisenberg-homogeneous norms on the Heisenberg groups given by $N_{α,A}((z,t)) = \left(|z|^α+ A |t|^{α/2}\right)^{1/α}$, for $α\ge 2$ and $A>0$. This natural family includes the canonical Cygan-Korányi norm, corresponding to $α=4$. 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 $α=2$ is the best possible, in all dimensions.

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