Variance of lattice point counting in some special shells in $\mathbb{R}^d$

TL;DR

This paper investigates the variability in counting lattice points within specific shells of certain convex domains in high-dimensional space, using Fourier analysis techniques.

Contribution

It introduces new variance estimates for lattice point counts in special shells, expanding understanding of lattice point distribution in convex domains.

Findings

Derived bounds on variance of lattice point counts

Applied Fourier transform estimates to convex domain indicators

Enhanced understanding of lattice point distribution in special shells

Abstract

We study the variance of the random variable that counts the number of lattice points in some shells generated by a special class of finite type domains in $\mathbb R^d$. The proof relies on estimates of the Fourier transform of indicator functions of convex domains.