A threshold phenomenon for embeddings of Euclidean snowflakes and impossibility of dimension reduction
Year: 2026
Type: preprint
Abstract: Fix $0<θ\leqslant 1$. We prove that if $1\leqslant p \leqslant 2/θ$, then the $θ$-snowflake of $\ell_2^k$, namely, $\mathbb{R}^k$ equipped with the metric $((x,y)\in \mathbb{R}^k\times \mathbb{R}^k)\mapsto \|x-y\|_2^θ$, embeds with distortion $O(1)$ into $\ell_p^m$ for some integer $m\lesssim_{p,θ}k$, which is optimal as $k\to \infty$, as seen by comparing dimensions. However, for $p$ larger than the sharp threshold $2/θ$ the following change in behavior occurs: If a $(1/\sqrt{k})$-dense subset of the Euclidean sphere $S^{k-1}$ embeds into $\ell_p^m$ with distortion $O(1)$, then necessarily $m\gtrsim_{p,θ}( k/\log k)^{pθ/2}$, which grows super-linearly in $k$ as $pθ/2>1$, and this dimension bound is optimal as $k\to \infty$ up to lower order factors. We deduce from this statement that if $2
Source: arXiv (Cornell University)
Authors: Naor, Assaf, Ren, Kevin
Language: English
Topic: Advanced Banach Space Theory
Subfield: Mathematical Physics
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7206149628