2018/07/17 by Abrams, Aaron, Landau, Henry, Landau, Zeph +3
#Combinatorics (math.CO) #FOS: Mathematics
paper · doi:10.48550/arxiv.1807.06495
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 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.