Topological objects associated to Steinerberger's 4-regular graphs 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 ?

 A: The Steinerberger article seems to be aimed at tests about randomness. The question you are raising, which is given as an initial observation in this article, is a topic in topological graph theory.

The following link should give you helpful references for your question: Reference for topological graph theory (research / problem-oriented) .

A: Here is a short Mathematica script that computes the graph and plots it with some standard function
f[n_] := Mod[n * Sqrt[2]//N, 1];

n = 200;
seq = f /@ Range[1,n];
map = PositionIndex[seq];
sort = map[#][[1]] & /@ (Sort@seq);

edge1 = Partition[Range[1,n], 2, 1] ~ Join ~ {{n,1}};
edge2 = Partition[sort, 2, 1] ~ Join ~ {{sort[[-1]], sort[[1]]}};
G = Graph[Join[edge1, edge2]]


GraphPlot3D[G, GraphLayout->"SpectralEmbedding"]


GraphPlot3D[G, GraphLayout->"SpringElectricalEmbedding"]


It seems to resemble some kind of genus 1 surface.

But seems to have nothing to do with $\sqrt2$. If I replace $\sqrt 2$ with $\pi$, the result still looks like a torus:

Apparently, all we need is that the number is irrational.
