AI 中文总结
该研究针对树覆盖问题，利用循环对称性将其下界改进为$\tilde{\beta}_k(n^{1/O(k^2)})$，缩小了与已知上界的指数差距，核心是通过循环对称性的灵活性得到关键结论。
AI 中文摘要
n点度量空间的树覆盖是指由k棵支配树组成的集合，使得每对距离至少被其中一棵树近似保持。目前已知的失真度一般上界为$\tilde{O}(n^{1/k})$。近期Chen、Tan和Xu（ITCS 2026，SICOMP 2026）利用拓扑方法证明了下界为$\tilde{\beta}_k(n^{1/2^{k-1}})$。我们将该下界改进为$\tilde{\beta}_k(n^{1/[k(p-1)]})=\tilde{\beta}_k(n^{1/O(k^2)})$，其中p是严格大于k的最小素数。由此，已知上界与下界指数间的差距从关于k的指数级缩小为$O(k)$倍。我们的关键发现是：之前方法中基于二进制标签的对跖对称性，与本文所用的循环对称性存在本质差异——在二进制场景中，每个联合标签有唯一的对跖伙伴，而$\tilde{\beta}_p^k$中的每个标签在每个坐标上都有多个不同的伙伴。这种灵活性使得仅需$k(p-1)$维的等变Borsuk-Ulam型定理就能生成两个相邻顶点，它们在所有k棵树中具有不同标签；随后的循环展开论证表明，它们在每棵树中的距离都很远。
英文摘要
A tree cover of an $n$-point metric space is a collection of $k$ dominating trees such that every pairwise distance is approximately preserved by at least one tree. The best known general upper bound on the distortion is $\widetilde{O}(n^{1/k})$. Recently, Chen, Tan, and Xu (ITCS 2026, SICOMP 2026) proved a lower bound of $Ω_k(n^{1/2^{k-1}})$ using a topological approach. We improve their lower bound to $Ω_k(n^{1/[k(p-1)]})=Ω_k(n^{1/O(k^2)})$, where $p$ is the smallest prime strictly larger than $k$. Thus, the gap between the known upper and lower exponents is reduced from exponential in $k$ to a factor of $O(k)$. Our key observation is a qualitative difference between the antipodal symmetry underlying the binary labels in the previous approach and the cyclic symmetry used here. In the binary setting, every joint label has a unique antipodal partner, whereas every label in $\mathbb{Z}_p^k$ has many partners that differ from it in every coordinate. This flexibility allows an equivariant Borsuk--Ulam-type theorem in only $k(p-1)$ dimensions to produce two nearby vertices with different labels in all $k$ trees. A cyclic unwinding argument then shows that they are far apart in every tree.
