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 ?

  • $\begingroup$ Isn't the paper saying that these surfaces are generated by the "van der Corpus sequence" rather than anything with $\sqrt 2$? $\endgroup$ – M. Winter Aug 8 at 9:09
  • $\begingroup$ For the plots shown, yes, but it is written that it also happens for these other sequences. I am intertested by any sequence producing a limit space with that construction. $\endgroup$ – Thomas Sauvaget Aug 8 at 9:56
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    $\begingroup$ @ThomasSauvaget I think Béart answer points to the right field. More specifically, I think the area where graph theory intersects with differential topology is relevant for you. Not directly related, but to give you an idea of this area and the machinery/techniques, look at the references in this post mathoverflow.net/q/368129/161328 $\endgroup$ – GraphX Aug 8 at 10:44
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    $\begingroup$ How do you embed the graph in $\mathbb{R}^3$? $\endgroup$ – Antoine Labelle Sep 4 at 20:19
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    $\begingroup$ @AntoineLabelle : thanks for the question, I now realise that these are actually plots of the graph in the plane (I'll edit the question). It still suggests that this is the projection of some higher dimensional manifold. $\endgroup$ – Thomas Sauvaget Sep 6 at 6:31

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) .

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  • $\begingroup$ Yes, I actually work in stats and was curious about the randomness test, but the geometric plot is hard to ignore. Thanks for the link. $\endgroup$ – Thomas Sauvaget Aug 8 at 10:59

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.

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  • $\begingroup$ Thanks for this code and plots, food for thought! $\endgroup$ – Thomas Sauvaget Aug 8 at 10:59

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