2017/04/28 by James Propp, Propp, James
Engineering · #Combinatorics (math.CO) #FOS: Mathematics #Optimization and Packing Problems
paper · pdf · doi:10.48550/arxiv.1704.08785
openalex publication_date 2017/04/28 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Every set of natural numbers determines a generating function convergent for q ∈ (-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 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.