Question

Is there a good asymptotic estimation for the number of integral points inside a $n$-sphere of a small radius ( less than $\sqrt{n}$)? It looks like a problem which has been studied, however, I cannot find any reference. Thanks for help!

Answer 1

Your question is treated in "Lattice points in high dimensional spheres" by J.E. Mazo and A.M. Odlyzko.

The article starts with

A general principle, dating back to Gauss, that is widely used in estimating the number of integer lattice points in nice sets $S$ in $\mathbb{R}^n$ is that this number equals the volume of $S$ with a small error term [4,5,13]. This approach is very useful, and can be proved to be rigorous, for example, if one considers the number of lattice points in sets $rT$, where the dimension $n$ is fixed, $T$ is a given nice set, and $r \to \infty$. However, we will show below that this principle fails completely if one considers the dimension $n \to \infty$, and the sets $S$ to be spheres of radii proportional to $\sqrt{n}$. That this general principle cannot be justified rigorously in this case is not surprising, since (i) the surface area of the sphere of radius $\sqrt{\alpha n }$ is larger than the volume by a factor of $\sqrt{n/\alpha}$, and (ii) the diameter of the unit cube is comparable to the diameter of the sphere. However, it is rather interesting that it is not just the proof, but the principle itself, that fails.

Now, I know that this is probably not the most recent paper on this problem but maybe you can search through the articles that reference it. See also Fedja's answer to this previous question.