Near-Optimal Embeddings of Constant-Dimensional Subspaces of $L^p$ into ${\mathrm{\ell }}_p^N$

For $d\ge 2$, $p\ge 1$ and $\epsilon >0$, let $N_p(d,\epsilon )$ be the smallest integer $N$ such that every $d$-dimensional subspace of $L^p[0,1]$ admits a linear embedding into ${\mathrm{\ell }}_p^N$ with distortion at most $1+\epsilon$. For fixed $d$ and $p$, the bound $$N_p(d,\epsilon )={\tilde{O}}_{d,p}\left({\epsilon }^{-2(d-1)\mathrm{/}(d+2p)}\right)$$ is established. For $p\mathrm{\notin }2\mathbb{Z}$, this is optimal up to logarithmic factors; for positive even integers $p$, isometric embeddings of dimension independent of $\epsilon$ are known. The stated upper bound was previously known only for integer $p$. The case of non-integral $p$ is handled by isolating the behaviour of $\mathrm{\mid }t{\mathrm{\mid }}^p$ near the origin, controlling this localized part by a discrepancy argument, and approximating the smooth remainder by polynomials.