I am given a hyperplane arrangement $\mathcal{H}_0$ in $\mathbb{R}^n$ and a function $\phi \colon \mathbb{R}^n \to \mathbb{Q}.$ I choose any enumeration on the set of primitive vectors (i.e. vectors whose entries are integers and have gcd = 1) $v_1, v_2, \cdots $. Then, I recursively form the hyperplane arrangments $\mathcal{H}_{i+1} = \mathcal{H}_i \cup \{l_{i+1}\}$ where $l_{i+1}$ is the line defined by $\langle v_{i+1}, x\rangle = -\phi(v_{i+1})$

I am interested in how the sum of the volumes of the bounded regions increase as I let this process go to infinity (i.e. $i \to \infty$). I would like something like a convergence criterion in terms of the function $\phi$.

Does anyone have any ideas? Or has a reference in mind which can help?