Long Sets of Lengths With Maximal Elasticity

Abstract

AbstractWe introduce a newinvariant describing the structure of sets of lengths in atomicmonoids and domains. For an atomic monoid H, let Δρ(H) be the set of all positive integers d that occur as differences of arbitrarily long arithmetical progressions contained in sets of lengths havingmaximal elasticity ρ(H). We study Δρ(H) for transfer Krull monoids of finite type (including commutative Krull domains with finite class group) with methods from additive combinatorics, and also for a class of weakly Krull domains (including orders in algebraic number fields) for which we use ideal theoretic methods.