  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 $\mathbb R^3$.
