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The main goal of this paper is to improve the result of Ostrovskii (2012) onthe finite determination of bilipschitz and coarse embeddability of locallyfinite metric spaces into Banach spaces.  There are two directions of the improvement:  (1) Substantial decrease of distortion (from about $3000$ to $3+\ep$) isachieved by replacing the barycentric gluing by the logarithmic spiral gluing.This decrease in the distortion is particularly important when the finitedetermination is applied to construction of embeddings.  (2) Simplification of the proof: a collection of tricks employed inOstrovskii (2012) is no longer needed, in addition to the logarithmic spiralgluing we use only the Brunel-Sucheston result on existence of spreadingmodels.

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