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](http://mathworld.wolfram.com/MinkowskiConvexBodyTheorem.html) (though this just guarantees 3 lattice points as far as I know).



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


**EDIT** For non-asymptotic bounds, see this [paper of Bentkus and Gotze](http://matwbn.icm.edu.pl/ksiazki/aa/aa80/aa8021.pdf) (Acta Arith. 1997), which has copious references also.


