Dimension-independent target-dimension version of the sharp Assouad distortion problem near theta=1

Establish whether the \(\theta\)-snowflake of every \(K\)-doubling metric space admits a Euclidean embedding with distortion \(O_K(1/\sqrt{1-\theta})\) as \(\theta\to1^-\), even when the target dimension is allowed to be an arbitrary function of \(K\) rather than restricted to \(O((\log K)/\theta)\).

Background

The conjectured sharp Assouad bounds predict distortion of order 1/1−θ1/\sqrt{1-\theta}, up to factors depending on KK, as θ\theta tends to one. The paper emphasizes that this behavior is unresolved even after relaxing the target-dimension requirement, making it a weaker but still open form of the conjecture.

References

Specifically, Conjecture~\ref{conj:sharp assouad} predicts that the distortion remains $O_K(1/\sqrt{1-\theta})$ as $\theta\to 1-$, which has been a longstanding open problem even if we relax the requirement that the target dimension is the optimal $O(\log K)$ by allowing it to be any function of $K$ whatsoever (see e.g.).

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