#####
Séminaire Lotharingien de Combinatoire, 80B.16 (2018), 11 pp.

# James Propp

# 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* -> 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 eventually periodic.
This implies an analogous result for packings in **N**. It is
conjectured that for all finite *D*
there is a unique maximal *D*-avoiding set.

Received: November 14, 2017.
Accepted: February 17, 2018.
Final version: April 1, 2018.

The following versions are available: