 
Summary: On the optimality of gluing over scales
Alex Jaffe
James R. Lee
Mohammad Moharrami
Abstract
We show that for every > 0, there exist npoint metric spaces (X, d) where every "scale"
admits a Euclidean embedding with distortion at most , but the whole space requires distortion
at least (
log n). This shows that the scalegluing lemma [Lee, SODA 2005] is tight, and
disproves a conjecture stated there. This matching upper bound was known to be tight at both
endpoints, i.e. when = (1) and = (log n), but nowhere in between.
More specifically, we exhibit npoint spaces with doubling constant requiring Euclidean dis
tortion (
log log n), which also shows that the technique of "measured descent" [Krauthgamer,
et. al., Geometric and Functional Analysis] is optimal. We extend this to Lp spaces with p > 1,
where one requires distortion at least ((log n)1/q
(log )11/q
) when q = max{p, 2}, a result
