# number of lattice points in an n-ball

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

• You mean to say that the lattice points are the images of some integer lattice under a linear transformation, and therefore $x\in\mathbb R^n$? So you might as well ask for integer lattice points in some $n$-dimensional paraboloid. Not that I think that formulation is any easier, mind, just thinking out loud. What kind of performance are you looking for? Would a scan conversion which gives an exact answer $m$ (i.e. there are $m$ points inside the ball) in time $O\left(m^{(n-1)/n}\right)$ be preferable to one which uses the bounding box for a very loose bound in $O(1)$? – MvG May 2 '14 at 13:05
• I was googling about the lattice points in an n-ball and found some papers about integer lattices (Gauss' circle problem) but in my problem the lattice is not integer. With $x\in \mathbb{R}$ I mean that the entries of the lattice point (vector $x$) are real numbers. I am not sure if the exact number of the lattice points inside an n-ball is solved but even a bound can be enough to proceed with my problem. – M.X May 2 '14 at 13:38
• What do you mean by “solved”? Sure you can compute that number, a brute force enumeration will yield the count eventually. On the other hand, a closed formula might be unrealistic. – MvG May 2 '14 at 14:08

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, $\omega_n R^n / |G|.$
• @jvnv looked at your posts, not sure what part of this could be a problem for you. The volume of the ball should be in any mathematical statistics book; the easiest method is induction by 2, even/odd dimension, using polar coordinates for the integral. Note that I am assuming $G$ expresses a basis of the lattice as its rows, so the Gram matrix is actually $G G^T.$ For anything in that area, try SPLAG by Conway and Sloane. Sphere Packing Lattices and Groups – Will Jagy Sep 25 '17 at 17:35