In standard rounding, we want to map each value $X$ in a large continuous space (e.g., $R$) to a nearby point $P$ from a discrete subset (e.g., $Z$). This process seems to be inherently discontinuous in the sense that two consecutive noisy measurements $X_1$ and $X_2$ of the same value may be extremely close to each other and yet they can be rounded to different points $P_1\ne P_2$, which is undesirable in many applications. In this paper we show how to make the rounding process perfectly continuous in the sense that it maps any pair of sufficiently close measurements to the same point. We call such a process consistent rounding, and make it possible by allowing a small amount of information about the first measurement $X_1$ to be unidirectionally communicated to and used by the rounding process of $X_2$. The fault tolerance of a consistent rounding scheme is defined by the maximum distance between pairs of measurements which guarantees that they are always rounded to the same point, and our goal is to study the possible tradeoffs between the amount of information provided and the achievable fault tolerance for various types of spaces. When the measurements $X_i$ are arbitrary vectors in $R^d$, we show that communicating $\log_2(d+1)$ bits of information is both sufficient and necessary (in the worst case) in order to achieve consistent rounding for some positive fault tolerance, and when d=3 we obtain a tight upper and lower asymptotic bound of $(0.561+o(1))k^{1/3}$ on the achievable fault tolerance when we reveal $\log_2(k)$ bits of information about how $X_1$ was rounded. We analyze the problem by considering the possible colored tilings of the space with $k$ available colors, and obtain our upper and lower bounds with a variety of mathematical techniques including isoperimetric inequalities, the Brunn-Minkowski theorem, sphere packing bounds, and \v{C}ech cohomology.

中文翻译：

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具有侧面信息的一致高维舍入

在标准舍入中，我们希望将大连续空间（例如，$R$）中的每个值 $X$ 映射到离散子集（例如，$Z$）中的附近点 $P$。这个过程似乎本质上是不连续的，因为两个连续的噪声测量值 $X_1$ 和 $X_2$ 可能彼此非常接近，但它们可以四舍五入到不同的点 $P_1\ne P_2$，这在许多应用中是不受欢迎的。在本文中，我们展示了如何使舍入过程完全连续，因为它将任何一对足够接近的测量值映射到同一点。我们将这种过程称为一致舍入，并通过允许关于第一个测量值 $X_1$ 的少量信息与 $X_2$ 的舍入过程单向通信并由其使用来实现。一致舍入方案的容错性由测量对之间的最大距离定义，这保证它们总是舍入到同一点，我们的目标是研究提供的信息量和可实现的容错之间的可能权衡适用于各种类型的空间。当测量值 $X_i$ 是 $R^d$ 中的任意向量时，我们表明传递 $\log_2(d+1)$ 位的信息是充分和必要的（在最坏的情况下），以便实现一致的舍入一些正容错，当 d=3 时，当我们揭示 $\log_2(k) 时，我们在可实现的容错上获得了严格的上下渐近界 $(0.561+o(1))k^{1/3}$ )$ 有关 $X_1$ 如何舍入的信息。