Lipschitz constant $\log{n}$ almost surely suffices for mapping $n$ grid points onto a cube

TL;DR

This paper demonstrates that, with high probability, the minimal Lipschitz constant needed to map a random set of points in a lattice onto a grid grows logarithmically with the grid size, providing new asymptotic bounds.

Contribution

It establishes almost sure upper bounds of order log n on the Lipschitz constant for random point configurations, advancing understanding of Lipschitz mappings in discrete settings.

Findings

Lipschitz constant grows as log n for random configurations

Almost sure asymptotic upper bounds are established

Provides probabilistic bounds on Lipschitz mappings

Abstract

Kalu\v{z}a, Kopeck\'a and the author have shown that the best Lipschitz constant for mappings taking a given $n^{d}$-element set in the integer lattice $\mathbb{Z}^{d}$, with $n\in \mathbb{N}$, surjectively to the regular $n$ times $n$ grid $\left\{1,\ldots,n\right\}^{d}$ may be arbitrarily large. However, there remain no known, non-trivial asymptotic bounds, either from above or below, on how this best Lipschitz constant grows with $n$. We approach this problem from a probabilistic point of view. More precisely, we consider the random configuration of $n^{d}$ points inside a given finite lattice and establish almost sure, asymptotic upper bounds of order $\log n$ on the best Lipschitz constant of mappings taking this set surjectively to the regular $n$ times $n$ grid $\left\{1,\ldots,n\right\}^{d}$.