Sharp Assouad embedding bounds for snowflakes of doubling metric spaces

Prove or disprove that, for every integer \(K\ge 3\) and every \(0<\theta<1\), the \(\theta\)-snowflake of every \(K\)-doubling metric space embeds into \(\ell_2^n\) with \(n\asymp (\log K)/\theta\) and distortion \(D\asymp (\log K)^\theta/\sqrt{1-\theta}\), with both orders optimal up to universal multiplicative constants.

Background

Assouad’s embedding theorem guarantees Euclidean embeddings of snowflakes of doubling metric spaces, but the conjecture seeks simultaneously optimal target dimension and sharp distortion dependence on the doubling constant and snowflake exponent. The paper provides partial upper and lower bounds and explains that the proposed estimates would in particular yield constant-distortion embeddings in the small-exponent regime.

References

The order of magnitude of the smallest possible distortion that one could achieve in Assouad's embedding theorem if one requires the target dimension to be of that smallest possible order of magnitude is the content of the following conjecture:

— A threshold phenomenon for embeddings of Euclidean snowflakes and impossibility of dimension reduction  (2609.01079 - Naor et al., 1 Sep 2026) in Conjecture 2.15, Section 2, “On the sharp Assouad problem”