Distribution and Moments of the Error Term in the Lattice Point Counting Problem for 3-Dimensional Cygan-Kor\'anyi Balls

TL;DR

This paper investigates the statistical distribution and moments of the error term in counting lattice points within 3D Cygan-Koranyi balls, revealing a limiting distribution and establishing the existence of all moments.

Contribution

It provides the first analysis of the error term's distribution and moments for lattice point counting in 3D Cygan-Koranyi balls, including decay estimates for the density.

Findings

Normalized error term has an absolutely continuous limiting distribution.

All moments of the normalized error term exist and match the moments of the limiting density.

Decay estimates for the density of the limiting distribution are established.

Abstract

We study fluctuations of the error term for the number of integer lattice points lying inside a 3-dimensional Cygan-Kor\'anyi ball of large radius. We prove that the error term, suitably normalized, has a limiting value distribution which is absolutely continuous, and we provide estimates for the decay rate of the corresponding density on the real line. In addition, we establish the existence of all moments for the normalized error term, and we prove that these are given by the moments of the corresponding density.