Consider the spherical shell (annulus) $$A(R,r) = \{ x \in \mathbb{R}^3 : R \leq | x|\leq R+r \}.$$ Think of the limit $R \to \infty$. Assume that $r$ depends on $R$ as $r(R) = R^{-\delta}$. We are interested in the distribution of lattice points in $A(R,r)$.

From results on the Gauss circle problem in three dimensions (see e.g. https://arxiv.org/abs/math/0410522) I know that the number of lattice points in the ball $B(R)$ is given by the volume of the ball, up to an error (the lattice point discrepancy) which is bounded by $\mathcal{O}(R^{\frac{42}{32}+\epsilon})$, for all $\epsilon > 0$. So by taking the difference we can obtain the number of lattice points in the annulus for $\delta >0$ not too large. We find that the number of lattice points is of order $R^{2}r$ (surface of the sphere times width of the shell).

In https://arxiv.org/abs/1204.0134 I found the reference to Duke & Golubeva-Fomenko, showing that the the lattice points exactly on the sphere are uniformly distributed, if $R^2 \neq 0,4,7 \mod 8$. (However there are only order $R^{1-\epsilon}$ lattice points exactly on the sphere.)

My question is: Is anything known about the distribution of lattice points in the thin spherical shell? For example, consider a spherical cap on $B(R)$ and 'fatten' it up to a radius $R+r$. We now have a segment of the annulus (if you want, the intersection of a cone with the annulus) and ask whether the number of lattice points in this segment is to leading order given by the area of the spherical cap times $r$. If yes, does this stay true if the solid angle defining the cap goes to zero (not too fast) as $R \to \infty$? If it is not true for all such segments, could I at least say that it is true for most segments, or for a specific sequence of radii R?

Another possible way of asking could be: consider summing a continuous function over $x/ |x|$, $x \in A(R,r)$ (and normalize by the number of points). Does the sum converge to the integral w.r.t. the uniform measure on the unit sphere?

Is anyone aware of results in this direction?