Lattice point counting statistics for 3-dimensional shrinking Cygan-Kor\'anyi spherical shells

TL;DR

This paper studies the statistical distribution of lattice point counting errors in 3D Cygan-Korányi spherical shells, showing they tend to a Gaussian distribution under certain conditions, with some cases yielding non-Gaussian limits.

Contribution

It establishes the limiting distribution of lattice point counting errors in 3D Cygan-Korányi shells, including Gaussian and non-Gaussian cases, under minimal assumptions on the gap width.

Findings

Normalized error term converges to a Gaussian distribution for slowly varying gap widths.

The distribution is moment-determinate and explicit moments are provided.

Constructed gap functions yield non-Gaussian absolutely continuous limiting distributions.

Abstract

Let $E(x;\omega)$ be the error term for the number of integer lattice points lying inside a $3$-dimensional Cygan-Kor\'anyi spherical shell of inner radius $x$ and gap width $\omega(x)>0$. Assuming that $\omega(x)\to0$ as $x\to\infty$, and that $\omega$ satisfies suitable regularity conditions, we prove that $E(x;\omega)$, properly normalized, has a limiting distribution. Moreover, we show that the corresponding distribution is moment-determinate, and we give a closed form expression for its moments. As a corollary, we deduce that the limiting distribution is the standard Gaussian measure whenever $\omega$ is slowly varying. We also construct gap width functions $\omega$, whose corresponding error term has a limiting distribution that is absolutely continuous with a non-Gaussian density.