The metric geometry of the Hamming cube and applications (Q309025)

Let \(\Delta_\infty\) be the metric space whose elements are all finite subsets of \(\mathbb{N}\), equipped with the cardinality of the symmetric difference metric. This metric space can be regarded as the infinite Hamming cube. Let \(\Delta_k\) be the subspace of \(\Delta_\infty\) consisting of all subsets of cardinality at most \(k\), and \(S_1\) be the subset consisting of Schreier sets, that is, sets for which the cardinality does not exceed the least element.NEWLINENEWLINEThe main goal of the paper is to study the distortions of bilipschitz embeddings of these metric spaces into Banach spaces \(C(K)\), where \(K\) is a countable metric compact. This goal can be regarded as the desire to interpolate between the results of \textit{S Banach} and \textit{S. Mazur} [Stud. Math. 4, 100--112 (1933; JFM 59.1075.01)]: Each separable metric space admits an isometric embedding into \(C[0,1]\); and the [\textit{N. J. Kalton} and \textit{G. Lancien}, Fundam. Math. 199, No. 3, 249--272 (2008; Zbl 1153.46047)] strengthening of the result of \textit{I. Aharoni} [Isr. J. Math. 19, 284--291 (1975; Zbl 0303.46012)]: Each separable metric space admits a bilipschitz embedding into \(c_0\) with distortion \(\leq 2\).NEWLINENEWLINEThe main results of the paper are as follows.NEWLINE{\parindent=7mm NEWLINE\begin{itemize}\item[1.] For every \(1\leq r\leq k\), there exists an embedding of \(\Delta_k\) into \(C([0,\omega^r])\) with distortion \(\leq\min\{k/r,2\}\). If \(r=k-1\), this bound is tight (to prove the lower bound, the authors use the Cantor-Bendixson index). NEWLINE\item[2.] The space \(S_1\) embeds isometrically into \(C([0,\omega^\omega])\). NEWLINE\item[3.] The space \(\Delta_\infty\) embeds almost isometrically into \(C([0,\omega^\omega])\). NEWLINE\item[4.] If \(\ell_1\) embeds almost isometrically into \(C(K)\), then \(\ell_1\) embeds almost isometrically into \(C(K^{(\alpha)})\) for all finite ordinals \(\alpha\), where \(K^{(\alpha)}\) is the Cantor-Bendixson derivative of order \(\alpha\). NEWLINENEWLINE\end{itemize}} NEWLINEThe last section is devoted to a discussion, including related open problems.