Sparse distribution of lattice points in annular regions

TL;DR

This paper investigates the sparse distribution of lattice points in large annular regions in two and three dimensions, establishing sharp thresholds for their spacing and demonstrating the existence of large gaps devoid of lattice points.

Contribution

It proves the existence of annuli with large radii and thickness where lattice points are sparsely distributed, with results sharp at the threshold s=1/4.

Findings

Existence of annuli with no close lattice points for s<1/4

Sharp threshold at s=1/4 for lattice point distribution

Extension of results to three-dimensional spherical shells

Abstract

This paper is inspired by Richards' work on large gaps between sums of two squares [10]. It is shown in [10] that there exist arbitrarily large values of $\lambda$ and $\mu$, where $\mu \geq C \log \lambda$, such that intervals $[\lambda, \,\lambda + \mu ]$ do not contain any sums of two squares. Geometrically, these gaps between sums of two squares correspond to annuli in $\mathbb R^2$ that do not contain any integer lattice points. A major objective of this paper is to investigate the sparse distribution of integer lattice points within annular regions in $\mathbb R^2$. Specifically, we establish the existence of annuli $\{x\in \mathbb R^2: \lambda \leq |x|^2 \leq \lambda + \kappa\}$ with arbitrarily large $\lambda$ and $\kappa \geq C \lambda^s$ for $0<s<\frac{1}{4}$, satisfying that any two integer lattice points within any one of these annuli must be sufficiently far apart. This result is sharp, as such a property ceases to hold at and beyond the threshold $s=\frac{1}{4}$. Furthermore, we extend our analysis to include the sparse distribution of lattice points in spherical shells in $\mathbb R^3$.