Very recently, in arXiv:2008.01153, Steinerberger has associated to any sequence $(x_n)_{n\in\mathbb{N}}$ of distinct real numbers a 4-regular graph.

In the case irrational multiples, like $x_n=n\sqrt{2} \pmod{1}$, the plots in $\mathbb{R}^2$ seem to show the projection of a certain genus-g surface (see page 2 of the preprint). [**edit**:06-sept-2020: I had written that these were plots in $\mathbb{R}^3$, which is actually not the case, apologies.]

is that indeed the case, i.e. does a limit shape as $n$ goes to infinity exist ? What type of literature (e.g. keywords, theorems) one should be looking at to establish it ?