# 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!

• All norms on $\mathbb{R}^n$ are equivalent, which means there are constants $c,C$ such that $c\|x\|_1 \leq \|x\|_2 \leq C\|x\|_1$ – Marcel Bischoff Jul 1 '17 at 23:07
• This is a standard application of Davenport's lemma, even though this particular result is much older. If you need more than what Davenport's lemma can give you, then likely more information is needed. – Stanley Yao Xiao Jul 1 '17 at 23:50
• @StanleyYaoXiao What is Davenport's lemma and what does it imply in this situation? – Johnny T. Jul 2 '17 at 0:31

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 $R:=R_1+\dots+R_n$.