We prove new results concerning the additive Galois module structure of wildly ramified non-abelian extensions $K/\mathbb{Q}$ with Galois group isomorphic to $A_4$, $S_4$, $A_5$, and dihedral groups of order $2p^n$ for certain prime powers $p^n$. In particular, when $K/\mathbb{Q}$ is a Galois extension with Galois group $G$ isomorphic to $A_4$, $S_4$ or $A_5$, we give necessary and sufficient conditions for the ring of integers $\mathcal{O}_{K}$ to be free over its associated order in the rational group algebra $\mathbb{Q}[G]$.

We introduce a newinvariant describing the structure of sets of lengths in atomicmonoids and domains. For an atomic monoid $H$, let ${{\Delta }_{\rho }}\left( H \right)$ be the set of all positive integers d that occur as differences of arbitrarily long arithmetical progressions contained in sets of lengths havingmaximal elasticity $\rho \left( H \right)$. We study ${{\Delta }_{\rho }}\left( H \right)$ 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.