Quantitative non-embeddability theorems and metric embeddings of slit carpets

Abstract

We study the bi-Lipschitz embedding problem for a class of metric spaces called slit carpets. First we show that the $n$th stage $\mathbb{M}_n$ of the standard slit carpet of Merenkov admits a bi-Lipschitz embedding into Euclidean space with distortion $ O(\sqrt{n})$. Then, we show a nearly sharp lower bound of $Ω\left(\sqrt{\frac{n}{\log(n)}}\right)$. This result quantifies the recent result by David and Eriksson-Bique, and thus gives a quantified answer to the question 8 in the paper by Heinonen and Semmes by showing that $\mathbb{M}_\infty$ does not bi-Lipschitz embed into Euclidean spaces. Then, we study the $L^1$ embeddability of the standard slit carpet. We show that the standard slit carpet has Lipschitz dimension $1$ in the sense of Cheeger and Kleiner, and consequently prove that it admits a bi-Lipschitz embedding into $ L^1 $. Third, we generalize the results in terms of targets and domains. First, we give a qualitative and Lebesgue differentiation based argument which shows that general slit carpets do not bi-Lipschitz embed into any Banach space with the RNP property. As a consequence, $\mathbb{M}_\infty$ does not bi-Lipschitz embed to $\ell_1$. We then consider carpets $\mathbb{M}^a$ where the relative sizes of slits decrease according to a sequence $a\in c_0$. We give a quantitative $β$-number based argument which shows that the carpets $\mathbb{M}^a$ do not bi-Lipschitz embed into Hilbert space if $a\not\in \ell_{1+ε}$.