AI 中文总结
该研究提出了将$L^p$中常维子空间近最优嵌入到$\ell_p^N$的理论界，针对不同$p$值给出了不同的最优嵌入维度。
AI 中文摘要
对于 \(d\geq 2\)，\(p\geq 1\) 以及 \(\epsilon > 0\)，设 \(N_{p}(d,\epsilon)\) 为最小整数 \(N\)，使得 \(L^{p}[0,1]\) 的每个 \(d\) 维子空间都能以至多 \(1 + \epsilon\) 的扭曲线性嵌入到 \(\ell_{p}^{N}\) 中。对于固定的 \(d\) 和 \(p\)，建立了界 \(N_{p}(d,\epsilon)=\widetilde{O}_{d,p}\!\left(\epsilon^{-2(d - 1)/(d + 2p)}\right)\)。对于 \(p\notin 2\mathbb{Z}\)，此界在对数因子范围内是最优的；对于正偶数 \(p\)，存在与 \(\epsilon\) 无关维度的等距嵌入。该上界之前仅对整数 \(p\) 已知。
英文摘要
For $d \geq 2$, $p \geq 1$ and $ε> 0$, let $N_p(d,ε)$ 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 $1 + ε$. For fixed $d\geq 2$ and $p\geq 1$, the bound \[ N_p(d,ε) \lesssim_{d,p} ε^{-2(d-1)/(d+2p)} \] 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 $ε$ 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.
CommentsRemoved logarithmic factors, achieving optimal target dimension (up to constant factors)
