Germ order for one-dimensional packings

TL;DR

This paper introduces a germ order for sets of natural numbers based on their generating functions, enabling comparison of sets and analysis of maximal D-avoiding sets, with applications to one-dimensional packing problems.

Contribution

It defines a new germ order for sets of natural numbers and characterizes maximal D-avoiding sets within this order, advancing understanding of packing problems.

Findings

Unique maximal D-avoiding sets in many cases

Germ order generalizes set cardinality and density

Applications to one-dimensional packing problems

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 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.