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 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 $\mathbb R^3$.