Bounding the number of lattice points inside an $n$-dimensional ellipsoid

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).

• What kind of bound do you expect? – Fan Zheng Apr 23 '15 at 20:53
• @FanZheng This might be naive since I'm still really new to all of this. That said, I am looking for a non-asymptotic upper and/or lower bound that is guaranteed to hold for any fixed $n$ (i.e. a lower bound that says you will have at least 10 lattice points in $E$ and at most 1000 lattice points in $E$). The bounds do not have to be related, and they can make use any of the quantities above (i.e. the volume of $E$). – Berk U. Apr 23 '15 at 21:12
• U Do you mind if the lower bound is negative if $r$ is positive but small? If you try to turn an asymptotic bound to an absolute one, this is likely to happen. – Fan Zheng Apr 23 '15 at 21:19
• @FanZheng No not at all. – Berk U. Apr 24 '15 at 18:37

Well, it depends on what you want to know. For homothetic images $t E$ of a fixed ellipsoid $E,$ the quantity is asymptotic to $t^n vol(E),$ where the error term is $O(t^{n-1}),$ with better bounds provable. But maybe you want to know something else?!