Counting points on lattices in inside a box- Geometry of numebrs Let $\Lambda$ be a lattice in $\mathbb{R}^n$ and let $|\mathbf{x}|$ denote the L2 norm. There is a fairly standard argument involving successive minima to obtain the estimate on $N(R)$ which is the number of points $\mathbf{x}$ on $\Lambda$ satisfying $|\mathbf{x}| \leq R$. 
Let $R_1, ..., R_n$ be positive real numbers. I was wondering if there was a way to estimate the number of points $\mathbf{x}$ on $\Lambda$ satisfying $|x_i| \leq R_i$ (for each $1\leq i \leq n$)?
PS I was thinking maybe one can apply a linear transformation to make the counting for a box with equals sides...
I am just not really sure what I can possibly do if I wanted such an estimate. Any comments or suggestions would be appreciated. Thank you very much!
 A: Let $\mathcal{S}\subset\mathbb{R}^n$ be a convex compact set lying in the closed ball of radius $R$ centered at the origin. Then
$$\left|\#(\mathcal{S}\cap\Lambda)-\frac{\mathrm{vol}(\mathcal{S})}{\det(\Lambda)}\right|\ll_n 1+\max_{1\leq m\leq n-1}\frac{R^m}{\lambda_1\dots\lambda_m},$$
where $\lambda_1\leq\dots\leq\lambda_n$ are the successive minima of $\Lambda$. In particular, the above bound applies for the box $\mathcal{S}=[-R_1,R_1]\times\dots\times[-R_n,R_n]$, upon choosing $R:=R_1+\dots+R_n$.
The above result is essentially Lemma 1 in Schmidt - Northcott's theorem on heights. II. The quadratic case (Acta Arith. 70 (1995), 343-375.), and it is based on Davenport's theorem. Schmidt's right hand side is slightly different, but the above version also follows by combining (2.1), (2.4), and the display after (2.4).
P.S. See also my answer to your earlier question.
A: This is not trivial (and Davenport's lemma, as in the comments, does not give great bounds). For more, see W. Schmidt's Diophantine Approximation (Chapter IV).
