Sparse distribution of lattice points in annular regions

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 λ and μ, where $\mu \ge C\log \lambda$, 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 $R^2$ 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 $R^2$. Specifically, we establish the existence of annuli $\{x\in R^2:\lambda \le |x{|}^2\le \lambda +\kappa \}$ with arbitrarily large λ and $\kappa \ge C{\lambda }^s$ 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 $R^3$.