论文标题
扁平托里的几乎最佳嵌入
Nearly Optimal Embeddings of Flat Tori
论文作者
论文摘要
我们表明，对于任何$ n $二维的晶格$ \ MATHCAL {l} \ subseteq \ Mathbb {r}^n $，torus $ \ mathbb {r}^n/\ mathcal {l} $可以嵌入$ o（\ o（可以嵌入$ o）中。这改善了$ o（n \ sqrt {\ log n}）$的先前最著名的上限，由Haviv and Regev（大约2010年）显示，并接近$ω（\ sqrt {n}）$的下限，这是由于Khot and Naor引起的（FOCS 2005，MATH。annal。Annal。2006）。
We show that for any $n$-dimensional lattice $\mathcal{L} \subseteq \mathbb{R}^n$, the torus $\mathbb{R}^n/\mathcal{L}$ can be embedded into Hilbert space with $O(\sqrt{n\log n})$ distortion. This improves the previously best known upper bound of $O(n\sqrt{\log n})$ shown by Haviv and Regev (APPROX 2010) and approaches the lower bound of $Ω(\sqrt{n})$ due to Khot and Naor (FOCS 2005, Math. Annal. 2006).
