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We study a lattice point counting problem for spheres arising from theHeisenberg groups. In particular, we prove an upper bound on the number ofpoints on and near large dilates of the unit spheres generated by theanisotropic norms $\|(z,t)\|_\alpha = ( |z|^\alpha +|t|^{\alpha/2})^{1/\alpha}$ for $\alpha\geq 2$. As a first step, we reduce ourcounting problem to one of bounding an energy integral. The primary newchallenges that arise are the presence of vanishing curvature and unevendilations. In the process, we establish bounds on the Fourier transform of thesurface measures arising from these norms. Further, we utilize the techniquesdeveloped here to estimate the number of lattice points in the intersection oftwo such surfaces.

Lattice Points Close to the Heisenberg Spheres