Lattice points in spheres

Question

Let $\mathbb{R}^n$ have the standard Euclidean metric and call a point $P = (x_1, \ldots,x_n)\in\mathbb{R}^n$ a lattice point if for all $i$, $x_i\in\mathbb{Z}$.

Allowing small number theoretic bazookas, it is not hard to see that given any $k\in\mathbb{N}$ there exists an open ball $B\subset\mathbb{R}^n$ containing exactly $k$ lattice points.

Can someone give an elementary proof of this or argue why this will have to rely on something non-elementary?

Answer 1

There is the classical geometrical fact, called the Steinhaus lattice point problem, that given a positive integer $n$, one may find a circle on the Euclidean plane surrounding exactly $n$ points of the integer lattice. There is a large literature on this and its generalizations, see for example the article

T. Kania, T. Kochanek, Steinhaus’ lattice-point problem for Banach spaces, J. Math. Anal. Appl. 446 (2017), 1219–1229.

and the references therein. As far as I can see, only some elementary arguments in metric spaces are needed.