Dvoretzky-type theorem for locally finite subsets of a Hilbert space

TL;DR

This paper proves that any locally finite subset of a Hilbert space can be nearly isometrically embedded into any infinite-dimensional Banach space, extending Dvoretzky's theorem to a broader setting.

Contribution

It introduces new techniques for embedding locally finite subsets of Hilbert spaces into arbitrary Banach spaces with controlled distortion.

Findings

Every locally finite subset of  admits a (1+psilon)-bilipschitz embedding into any infinite-dimensional Banach space.

Constructs controlled embeddings of finite-dimensional Euclidean spaces into sums of Banach spaces.

Provides methods for near-isometric embeddings that preserve local structure in Banach spaces.

Abstract

The main result of the paper: Given any $\varepsilon>0$, every locally finite subset of $\ell_2$ admits a $(1+\varepsilon)$-bilipschitz embedding into an arbitrary infinite-dimensional Banach space. The result is based on two results which are of independent interest: (1) A direct sum of two finite-dimensional Euclidean spaces contains a sub-sum of a controlled dimension which is $\varepsilon$-close to a direct sum with respect to a $1$-unconditional basis in a two-dimensional space. (2) For any finite-dimensional Banach space $Y$ and its direct sum $X$ with itself with respect to a $1$-unconditional basis in a two-dimensional space, there exists a $(1+\varepsilon)$-bilipschitz embedding of $Y$ into $X$ which on a small ball coincides with the identity map onto the first summand and on a complement of a large ball coincides with the identity map onto the second summand.