One-Dimensional Packing: Maximality Implies Rationality

Abstract

Every set of natural numbers determines a generating function convergent for $q \in (-1,1)$ whose behavior as $q \rightarrow 1^-$ determines a germ. These germs admit a natural partial ordering that can be used to compare sizes of 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 $D$. It is shown that any $D$-avoiding set that is maximal in the class of $D$-avoiding sets (with respect to germ-ordering) is ultimately periodic. This implies an analogous result for packings. It is conjectured that for all $D$ there is a unique maximal $D$-avoiding set, and that its germ is appreciably larger than the germs of all other $D$-avoiding sets.