On Connectedness of Discretized Sets

摘要

Constructing a discretization of a given set is a major problem in various theoretical and applied disciplines. An offset discretization of a set X is obtained by taking the integer points inside a closed neighborhood of X of a certain radius. In this note we determine a minimum threshold for the offset radius, beyond which the discretization of an arbitrary (possibly disconnected) set is always connected. The results hold for a broad class of disconnected subsets of R～n, and generalize several previous results.