# Long sets of lengths with maximal elasticity

## TL;DR

This paper introduces a new invariant to analyze the structure of sets of lengths in atomic monoids, using additive combinatorics and ideal theory to study their properties in various algebraic contexts.

## Contribution

It defines the invariant $ ext{Delta}_
ho(H)$ for atomic monoids and explores its properties in transfer Krull monoids and weakly Krull domains, advancing the understanding of factorizations.

## Key findings

- Characterization of $ ext{Delta}_
ho(H)$ for transfer Krull monoids.
- Analysis of $ ext{Delta}_
ho(H)$ in weakly Krull domains.
- Application of additive combinatorics and ideal theory methods.

## Abstract

We introduce a new invariant describing the structure of sets of lengths in atomic monoids and domains. For an atomic monoid $H$, let $\Delta_{\rho} (H)$ be the set of all positive integers $d$ which occur as differences of arbitrarily long arithmetical progressions contained in sets of lengths having maximal elasticity $\rho (H)$. We study $\Delta_{\rho} (H)$ 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.