On the distribution of lattice points in thin annuli

TL;DR

This paper demonstrates that the count of lattice points in a thin annulus follows a Gaussian distribution when the annulus width decreases slowly relative to the increasing inner radius.

Contribution

It establishes a probabilistic distribution result for lattice points in thin annuli, extending understanding of their statistical behavior in geometric number theory.

Findings

Lattice point counts in thin annuli tend to a Gaussian distribution.

The distribution holds when the annulus width decreases slowly with radius.

Provides new insights into the statistical properties of lattice points in geometric regions.

Abstract

We show that the number of lattice points lying in a thin annulus has a Gaussian value distribution if the width of the annulus tends to zero sufficiently slowly as we increase the inner radius.