Metric embeddings of cubes into dense subsets of cubes

TL;DR

This paper investigates the conditions under which dense subsets of high-dimensional hypercubes contain metric embeddings of smaller cubes, providing bounds for various embedding types and exploring geometric curvature implications.

Contribution

It introduces new bounds for embedding smaller cubes into dense subsets of hypercubes under different metric constraints and applies these results to curvature and coloring problems.

Findings

Bound for bi-Lipschitz embeddings: N = O(ε^{-2} log(1/δ) k^3)

Bound for isometric embeddings with arbitrary rescaling: N = log(1/δ) e^{Ω(k)}

Bound for isometric embeddings with bounded rescaling: N = exp[log(1/δ) e^{Θ(k)}]

Abstract

Fix $k \in \mathbb{N}$ and $0 < \delta < 1$. We study how large $N$ must be so that every $\delta$-dense subset $\mathcal{D} \subset \{0,1\}^N$ (meaning $|\mathcal{D}| \geq \delta 2^N$) contains the image of a metric embedding $f: \{0,1\}^k \to \mathcal{D}$. We study three variants. For a $(1+\varepsilon)$-bi-Lipschitz map $f$ with fixed $\varepsilon > 0$, we show $N = O(\varepsilon^{-2} \log(1/\delta) k^3)$. For an isometric map with arbitrary rescaling (undistorted), we show $N = \log(1/\delta) e^{\Omega(k)}$ and conjecture $N = \log(1/\delta) e^{O(k)}$. For an isometric map with bounded rescaling we show $N = \exp[\log(1/\delta) e^{\Theta(k)}]$. As a geometric application, we obtain a nonpositive Alexandrov curvature counterpart to the work of Bartal-Linial-Mendel-Naor on the nonlinear Dvoretzky problem. It is known that any subset of $\{0,1\}^N$ embedding with bi-Lipschitz distortion $< \alpha$ into a metric space of nonnegative Alexandrov curvature must satisfy $|\mathcal{D}| \lesssim 2^{N(1-\Omega(\alpha^{-2}))}$. Work of Gromov and Kondo shows that this approach does not extend to CAT(0) targets. We prove that for every $N \gtrsim \alpha^6 \geq 1$, any $\mathcal{D} \subset \{0,1\}^N$ embedding with distortion $< \alpha$ into a CAT(0) space must satisfy $|\mathcal{D}| \lesssim 2^{N(1-\Omega(\alpha^{-4}))}$, via a completely different approach. Similar results hold for targets of nontrivial Enflo type. Finally, we prove the density analogue of a coloring theorem of Rodl-Sales: we give bounds for $(1+\varepsilon)$-bi-Lipschitz embeddings of the path $\{1,\ldots,k\}$ into dense subsets of $\{1,\ldots,N\}$ (improving a bound of Dumitrescu), and prove similar bounds for binary tree metrics.