Finite determination for embeddings into Banach spaces: New proof, low distortion

The main goal of this paper is to improve the result of M. I. Ostrovskii [Embeddability of locally finite metric spaces into Banach spaces is finitely determined, Proc. Amer. Math. Soc. 140 (2012) 2721–2730.] on the finite determination of bilipschitz and coarse embeddability of locally finite metric spaces into Banach spaces. There are two directions of the improvement: (1) Substantial decrease of distortion (from about [Formula: see text] to [Formula: see text]) is achieved by replacing the barycentric gluing by the logarithmic spiral gluing. This decrease in the distortion is particularly important when the finite determination is applied to construction of embeddings. (2) Simplification of the proof: a collection of tricks employed in M. I. Ostrovskii [Embeddability of locally finite metric spaces into Banach spaces is finitely determined, Proc. Amer. Math. Soc. 140 (2012) 2721–2730.] is no longer needed; in addition to the logarithmic spiral gluing we use only a version of the Brunel–Sucheston argument used to prove the existence of spreading models.