number of lattice points in an n-ball

Question

I have faced a problem in my work and I will appreciate any hint/reference as I am not much into the lattice problems.

Assume an n-dimensional lattice $\Lambda_n$ with generator matrix $G$. Note that lattice points are not necessarily integer, i.e., $x\in \mathbb{R}$ where $x$ is a lattice point. Is there a way to count/estimate/bound the number of lattice points inside and on an n-ball?

any hint or reference to appropriate literature is appreciated

Answer 1

The $n$-volume of the fundamental parallelotope is the absolute value of the determinant of $G.$

The $n$-volume of the ball of radius $1$ is, in shorthand, $\pi^{n/2}/ (n/2)!,$ or $$ \omega_n = \frac{\pi^{n/2}}{\Gamma \left( 1 + \frac{ n}{2} \right)}. $$ The volume of the ball of radius $R$ is $\omega_n R^n. $

So, there is your estimate of the count, $\omega_n R^n / |G|.$