Basically, I'm looking for ways to multiply elements of $\mathbb{R}^n$ that allow me to count divisors in $\mathbb{Z}^n$.

For every positive integer $n$, I'm looking for an algebra structure on $\mathbb{R}^n$ such that

- Given $y,z \in \mathbb{Z}^n$ with $z$ non-zero, $x y = z$ has at most $c$ solutions $x \in \mathbb{Z}^n$, where $c$ is a constant independent of $y$ and $z$.
- Given $z \in \mathbb{Z}^{n}$ with $z$ non-zero, $$\# \{ (x,y) \in \mathbb{Z}^{2n} : xy = z \} = \| z \|^{o(1)},$$ where $\| \cdot \|$ is the usual Euclidean norm on $\mathbb{R}^n$.

Can $\mathbb{R}^n$ be given such an algebra structure?

Examples: If $n = 2$, use $\mathbb{C}$. If $n=4$, maybe use the quaternions. It's not exactly clear what the divisor bound is for the quaternions.

I know that the only finite-dimensional real division algebras are the real numbers, the complex numbers, the quaternions, and the octonions.