Sparse distribution of lattice points in annular regions

Abstract

This paper is inspired by Richards' work on large gaps between sums of two squares [10]. It is demonstrated 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. The primary 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 values of $\lambda$ and $\kappa$, where $\kappa \geq C \lambda^s$ with $0<s<\frac{1}{4}$, satisfying that any two integer lattice points within any one of these annuli must be sufficiently far apart. Furthermore, we extend our analysis to include the sparse distribution of lattice points in spherical shells in $\mathbb R^3$.Comment: 13 page