I am wondering if it is possible to produce an upper and/or lower bound on the number of integer lattice points that lie **inside** an $n$-dimensional ellipse.

That is, given an $n$-dimensional ellipsoid with a center at $c \in \mathbb{R}^n$: $$E(A,c,r) := \Big\{x \in \mathbb{R}^n ~\Big| ~(x-c)^T A (x-c) \leq r\Big\},$$

does there exist an upper and/or lower bound for the quantity:

$$\#(E(A,c,r)) := \big|E(A,c,r) \cap \mathbb{Z}^n \big|$$

If so, could someone cite the result / direct me towards the appropriate reference?

I'd appreciate any guidance. I'm really new to this area (and very unfamiliar with the terms). The only relevant result that I know is Minkowski's Convex Body theorem (though this just guarantees 3 lattice points as far as I know).