Catrina, FlorinOstrovska, SofiyaOstrovskii, Mikhail I.2025-10-062025-10-0620250373-09561777-531010.5802/aif.36722-s2.0-105014409907https://doi.org/10.5802/aif.3672https://hdl.handle.net/20.500.14411/10853The main result of the paper: Given any epsilon > 0, every locally finite subset of l(2) admits a (1 + epsilon)-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 epsilon-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 + epsilon)-bilipschitz embedding of Y into X which on a small ball coincides with the identity map onto the first summand and on the complement of a large ball coincides with the identity map onto the second summand.eninfo:eu-repo/semantics/openAccessBilipschitz EmbeddingDvoretzky TheoremFinite-Dimensional DecompositionUnconditional BasisArticle