I wish to find a specific discretization of $\mathbb{R}^n$ in such a way that the subsets obtained form hyperplanes of codimension 1. Let me explain:
If we consider a prime number $p$ then $\mathbb{Z}_p\times\mathbb{Z}_p$ is a product of cyclic groups of order $p^2$ and thus there are $p+1$ nontrivial proper subgroups, each one of them of order $p$, which are $$H_i=\{(k,l) \in \mathbb{Z}_p\times \mathbb{Z}_p : \; il\equiv k \; {\mbox{mod}}\; p\}, \: i=0, 1, ...,p-1, \quad\text{ and }\quad H_p=\{(k,0), \; k\in \mathbb{Z}_p\}.$$
Geometrically, they represent $(p+1)$ lines covering $\mathbb{Z}_p\times \mathbb{Z}_p$ and all going through the origin.
Now my goal is to expand this to $\mathbb{Z}_p^n$. (Note that i haven't checked any of what i claim in what follows. This is rather an "intuitive guess"). Following the same reasoning, there are $p+1$ nontrivial proper subgroups of codimension $1$ each of them of order $p^{n-1}$, which are $$H_i=\left\{z\in\mathbb{Z}^n_p\big| \exists l,k\in\{1,\dots , n\}:iz_l\equiv z_k\mod p\right\} \text{ for }i\in\{0,1,\dots,p-1\} \quad\text{ and }H_p=\left\{z\in\mathbb{Z}^n_p\big| \forall k\in\{1,\dots,n-1\}, z_k\in\mathbb{Z}_p\text{ and }z_n\equiv 0\mod p\right\}$$
As I said before this is just a guess (which is highly likely to be wrong) and I wish to do such a construction more precisely but I haven't find good sources for such topic.
Do you have any specific recommendations? Are my guesses correct?
Have a great day!