Lattice Points Close to the Heisenberg Spheres

TL;DR

This paper investigates lattice point counts near Heisenberg sphere dilates defined by anisotropic norms, establishing bounds despite challenges like vanishing curvature and uneven dilations, and extends results to intersections of such surfaces.

Contribution

It introduces new bounds on lattice points near anisotropic Heisenberg spheres, addressing curvature and dilation issues, and develops Fourier analysis techniques for these norms.

Findings

Bound on lattice points near Heisenberg spheres

Bounds on Fourier transforms of surface measures

Estimates for lattice points in intersections of surfaces

Abstract

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