Optimal conditions for connectedness of discretized sets

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 a disconnected set is always connected. The results hold for a broad class of disconnected and unbounded subsets of $R^n$, and generalize several previous results. Algorithmic aspects and possible applications are briefly discussed.