Germ order for one-dimensional packings

Every set of natural numbers determines a generating function convergent for \( q \in (-1, 1) \) whose behavior as \( q \to 1^- \) determines a germ. These germs admit a natural partial ordering that can be used to compare sets of natural numbers in a manner that generalizes both cardinality of finite sets and density of infinite sets. For any finite set \( D \) of positive integers, call a set \( S \) “\( D \)-avoiding” if no two elements of \( S \) differ by an element of \( D \). We study the problem of determining, for fixed \( D \), all \( D \)- avoiding sets that are maximal in the germ order. In many cases, we can show that there is exactly one such set. We apply this to the study of one-dimensional packing problems.