A partition $\mathcal{P}$ of $\mathbb{R}^d$ is called a $(k,\varepsilon)$-secluded partition if, for every $\vec{p} \in \mathbb{R}^d$, the ball $\overline{B}_{\infty}(\varepsilon, \vec{p})$ intersects at most $k$ members of $\mathcal{P}$. A goal in designing such secluded partitions is to minimize $k$ while making $\varepsilon$ as large as possible. This partition problem has connections to a diverse range of topics, including deterministic rounding schemes, pseudodeterminism, replicability, as well as Sperner/KKM-type results. In this work, we establish near-optimal relationships between $k$ and $\varepsilon$. We show that, for any bounded measure partitions and for any $d\geq 1$, it must be that $k\geq(1+2\varepsilon)^d$. Thus, when $k=k(d)$ is restricted to ${\rm poly}(d)$, it follows that $\varepsilon=\varepsilon(d)\in O\left(\frac{\ln d}{d}\right)$. This bound is tight up to log factors, as it is known that there exist secluded partitions with $k(d)=d+1$ and $\varepsilon(d)=\frac{1}{2d}$. We also provide new constructions of secluded partitions that work for a broad spectrum of $k(d)$ and $\varepsilon(d)$ parameters. Specifically, we prove that, for any $f:\mathbb{N}\rightarrow\mathbb{N}$, there is a secluded partition with $k(d)=(f(d)+1)^{\lceil\frac{d}{f(d)}\rceil}$ and $\varepsilon(d)=\frac{1}{2f(d)}$. These new partitions are optimal up to $O(\log d)$ factors for various choices of $k(d)$ and $\varepsilon(d)$. Based on the lower bound result, we establish a new neighborhood version of Sperner's lemma over hypercubes, which is of independent interest. In addition, we prove a no-free-lunch theorem about the limitations of rounding schemes in the context of pseudodeterministic/replicable algorithms.

翻译：$\mathbb{R}^d$的一个划分$\mathcal{P}$被称为$(k,\varepsilon)$-隔离划分，如果对于每个$\vec{p} \in \mathbb{R}^d$，球$\overline{B}_{\infty}(\varepsilon, \vec{p})$至多与$\mathcal{P}$的$k$个成员相交。设计此类隔离划分的目标是最小化$k$，同时尽可能最大化$\varepsilon$。这一划分问题与多种主题相关，包括确定性舍入方案、伪确定性、可复制性以及Sperner/KKM型结果。在本文中，我们建立了$k$与$\varepsilon$之间的近最优关系。我们证明，对于任意有界测度划分及任意$d\geq 1$，必有$k\geq(1+2\varepsilon)^d$。因此，当$k=k(d)$限制为${\rm poly}(d)$时，可得$\varepsilon=\varepsilon(d)\in O\left(\frac{\ln d}{d}\right)$。该界在对数因子意义下是紧的，已知存在$k(d)=d+1$且$\varepsilon(d)=\frac{1}{2d}$的隔离划分。我们还提供了适用于广泛$k(d)$和$\varepsilon(d)$参数范围的新型隔离划分构造。具体地，我们证明，对于任意$f:\mathbb{N}\rightarrow\mathbb{N}$，存在隔离划分满足$k(d)=(f(d)+1)^{\lceil\frac{d}{f(d)}\rceil}$且$\varepsilon(d)=\frac{1}{2f(d)}$。这些新划分在$k(d)$和$\varepsilon(d)$的各种选择下，在$O(\log d)$因子内达到最优。基于下界结果，我们在超立方体上建立了一个独立意义的新型邻域版Sperner引理。此外，我们证明了关于伪确定性/可复制算法中舍入方案局限性的无免费午餐定理。