Chirotopes of Random Points in Space are Realizable on a Small Integer Grid

TL;DR

This paper demonstrates that random point sets in high-dimensional space can be efficiently encoded on small integer grids without altering their chirotopes, contrasting with worst-case scenarios requiring exponentially small grid steps.

Contribution

It proves that with high probability, random points in convex domains can be rounded to small integer grids preserving their chirotopes, extending previous planar results to higher dimensions.

Findings

Random points can be rounded to small integer grids without changing chirotopes.

Chirotopes of random point sets can be encoded with O(n log n) bits.

Contrast with worst-case requiring exponentially small grid steps.

Abstract

We prove that with high probability, a uniform sample of $n$ points in a convex domain in $\mathbb{R}^d$ can be rounded to points on a grid of step size proportional to $1/n^{d+1+\epsilon}$ without changing the underlying chirotope (oriented matroid). Therefore, chirotopes of random point sets can be encoded with $O(n\log n)$ bits. This is in stark contrast to the worst case, where the grid may be forced to have step size $1/2^{2^{\Omega(n)}}$ even for $d=2$. This result is a high-dimensional generalization of previous results on order types of random planar point sets due to Fabila-Monroy and Huemer (2017) and Devillers, Duchon, Glisse, and Goaoc (2018).