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!



Counting points on lattices in inside a box- Geometry of numebrs
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](https://mathoverflow.net/questions/258412/counting-number-of-points-on-a-lattice-in-a-hypercube).


