Long Sets of Lengths With Maximal Elasticity. (1st December 2018) - Geroldinger, Alfred; Zhong, Qinghai
Abstract: We introduce a newinvariant describing the structure of sets of lengths in atomicmonoids and domains. For an atomic monoid $H$, let ${{Δ }_{ρ }}\left( H \right)$ be the set of all positive integers d that occur as differences of arbitrarily long arithmetical progressions contained in sets of lengths havingmaximal elasticity $ρ \left( H \right)$ . We study ${{Δ }_{ρ }}\left( H \right)$ 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.