Lattice point counting problems on step-two nilpotent Lie groups

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}_{α,M}(x, t)=\left(|M_1x|^α+ |M_2t|^{α/ 2}\right)^{1 / α}$ adapted to the dilation structure, wh