A major research area in discrete geometry is to consider the best way to partition the $d$-dimensional Euclidean space $\mathbb{R}^d$ under various quality criteria. In this paper we introduce a new type of space partitioning that is motivated by the problem of rounding noisy measurements from the continuous space $\mathbb{R}^d$ to a discrete subset of representative values. Specifically, we study partitions of $\mathbb{R}^d$ into bounded-size tiles colored by one of $k$ colors, such that tiles of the same color have a distance of at least $t$ from each other. Such tilings allow for \emph{error-resilient} rounding, as two points of the same color and distance less than $t$ from each other are guaranteed to belong to the same tile, and thus, to be rounded to the same point. The main problem we study in this paper is characterizing the achievable tradeoffs between the number of colors $k$ and the distance $t$, for various dimensions $d$. On the qualitative side, we show that in $\mathbb{R}^d$, using $k=d+1$ colors is both sufficient and necessary to achieve $t>0$. On the quantitative side, we achieve numerous upper and lower bounds on $t$ as a function of $k$. In particular, for $d=3,4,8,24$, we obtain sharp asymptotic bounds on $t$, as $k \to \infty$. We obtain our results with a variety of techniques including isoperimetric inequalities, the Brunn-Minkowski theorem, sphere packing bounds, Bapat's connector-free lemma, and \v{C}ech cohomology.

翻译：离散几何中的一个主要研究领域是考虑在多种质量准则下划分 $d$ 维欧几里得空间 $\mathbb{R}^d$ 的最佳方式。本文引入一种新型的空间划分，其动机源于将连续空间 $\mathbb{R}^d$ 中的噪声测量值舍入到离散代表值集合的问题。具体而言，我们研究将 $\mathbb{R}^d$ 划分为有界大小的瓦片，并用 $k$ 种颜色之一着色，使得相同颜色的瓦片彼此之间的距离至少为 $t$。这种平铺允许进行**误差弹性**舍入，因为相同颜色且彼此距离小于 $t$ 的两个点保证属于同一瓦片，从而被舍入到同一点。本文研究的主要问题是刻画在不同维度 $d$ 下，颜色数 $k$ 与距离 $t$ 之间可实现的权衡关系。在定性方面，我们证明在 $\mathbb{R}^d$ 中，使用 $k=d+1$ 种颜色既是实现 $t>0$ 的充分条件，也是必要条件。在定量方面，我们获得了关于 $t$ 作为 $k$ 的函数的多个上界和下界。特别地，对于 $d=3,4,8,24$，当 $k \to \infty$ 时，我们得到了 $t$ 的尖锐渐近界。我们运用多种技术获得了这些结果，包括等周不等式、Brunn-Minkowski 定理、球堆积界、Bapat 的无连接引理以及 \v{C}ech 上同调。