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!