Geodesic
One-Dimensional Packing: Maximality Implies Rationality
Séminaire lotharingien de combinatoire, 80B (2018)

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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. 

@article{SLC_2018_80B_a15,
     author = {James Propp},
     title = {One-Dimensional {Packing:} {Maximality} {Implies} {Rationality}},
     journal = {S\'eminaire lotharingien de combinatoire},
     publisher = {mathdoc},
     volume = {80B},
     year = {2018},
     url = {http://geodesic.mathdoc.fr/item/SLC_2018_80B_a15/}
}
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AU  - James Propp
TI  - One-Dimensional Packing: Maximality Implies Rationality
JO  - Séminaire lotharingien de combinatoire
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UR  - http://geodesic.mathdoc.fr/item/SLC_2018_80B_a15/
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James Propp. One-Dimensional Packing: Maximality Implies Rationality. Séminaire lotharingien de combinatoire, 80B (2018). http://geodesic.mathdoc.fr/item/SLC_2018_80B_a15/