An offset discretization of a set \(X \subset {\mathbb {R}}^n\) is obtained by taking the integer points inside a closed neighborhood of X of a certain radius. In this work we determine a minimum threshold for the offset radius, beyond which the discretization of an arbitrary (possibly disconnected) set is always connected. The obtained results hold for a broad class of disconnected subsets of \({\mathbb {R}}^n\) and generalize several previous results. We also extend our results to infinite discretizations of unbounded subsets of \({\mathbb {R}}^n\) and consider certain algorithmic aspects. The obtained results can be applied to component topology preservation as well as extracting geometric features and connectivity control of very large object discretizations.

集合\（X \ subset {\ mathbb {R}} ^ n \）的偏移离散化是通过将某个半径为X的封闭邻域内的整数点获得的。在这项工作中，我们确定了偏移半径的最小阈值，在该阈值之上始终连接任意（可能是断开的）集合的离散化。所获得的结果适用于\（{\ mathbb {R}} ^ n \）的一类断开连接的子集，并且可以概括先前的几个结果。我们还将结果扩展到\（{{mathbb {R}} ^ n \）的无穷子集的无限离散化并考虑某些算法方面。所获得的结果可以应用于组件拓扑的保留，以及提取几何特征和非常大的对象离散化的连接控制。