Optimal Embeddings of Constant-Dimensional Subspaces of $L^p$ into $\ell_p^N$

Abstract

For $d \geq 2$, $p \geq 1$ 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 $\ell_p^N$ with distortion at most $1 + \epsilon$. For fixed $d\geq 2$ and $p\geq 1$, the bound \[ N_p(d,\epsilon) \lesssim_{d,p} \epsilon^{-2(d-1)/(d+2p)} \] is established. For $p \notin 2\mathbb{Z}$, this matches the known lower bound up to constant factors. For odd integers $p$, previous upper bounds with this exponent incurred additional logarithmic factors, except in the logarithm-free case $p = 1$; for non-integral $p$, no upper bound with this exponent was previously known. For even integers $p$, isometric embeddings of dimension independent of $\epsilon$ are known. For $p \notin 2\mathbb{Z}$, the proof approximates $|t|^p$ by a polynomial with a remainder of small total variation. The polynomial part contributes no error, while the error from the remainder is controlled by an integrated equatorial-band discrepancy estimate.