Optimal conditions for connectedness of discretized sets

TL;DR

This paper establishes the minimum offset radius needed to ensure the discretization of any disconnected set in Euclidean space becomes connected, broadening previous results and applicable to various sets.

Contribution

It determines a universal threshold for offset radius that guarantees connectedness of discretized sets, generalizing earlier findings and applicable to broad classes of sets.

Findings

Identifies a minimum offset radius for connected discretizations

Applicable to broad classes of disconnected and unbounded sets

Provides insights into algorithmic and practical applications

Abstract

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.