Naor's sublinear-dimension question for finite subsets of Lp, 1 < p < infinity, p not 2

Theorem 1.1: for fixed $1<p<\infty$, $p\ne2$, and $D>1$, every $n$-point subset of any $L_p$ embeds with distortion $D$ into $\ell_p^d$ (same exponent, coordinate target, nonlinear map) with $d\le\exp(C_{p,D}(\log n)^{\gamma(p)})=n^{o(1)}$, where $\gamma(p)=2-p$ for $p<2$ and $1-2/p$ for $p>2$; exact embeddings need dimension of order $n^2$. This answers the $o(n)$ question for every $p\in(1,\infty)\setminus\{2\}$. Not shown: $p=1$; Naor's Question 13 itself ($O(\log n)$ dimension in some normed space), which Naor-Ren's 2026 lower bound already excludes for $\ell_p^d$ targets when $p>2$; optimal exponents; an efficient algorithm.

Theorem 1.1: for fixed $1<p<\infty$, $p\ne2$, and $D>1$, every $n$-point subset of any $L_p$ embeds with distortion $D$ into $\ell_p^d$ (same exponent, coordinate target, nonlinear map) with $d\le\exp(C_{p,D}(\log n)^{\gamma(p)})=n^{o(1)}$, where $\gamma(p)=2-p$ for $p<2$ and $1-2/p$ for $p>2$; exact embeddings need dimension of order $n^2$. This answers the $o(n)$ question for every $p\in(1,\infty)\setminus\{2\}$. Not shown: $p=1$; Naor's Question 13 itself ($O(\log n)$ dimension in some normed space), which Naor-Ren's 2026 lower bound already excludes for $\ell_p^d$ targets when $p>2$; optimal exponents; an efficient algorithm.