Almost lattices are an instance of aperiodic order. Usually, they show up in the form of model sets in the sense of Meyer. More recently, weak model sets have also gained attention due to their significance in number theory. Although no longer almost lattices, they are limits of sequences of proper model sets. In Project A2, the principal goal is to find fine, yet computationally accessible invariants for aperiodic tiling spaces (via characteristic point sets) and for aperiodic shift spaces, both viewed as dynamical systems. Important examples come from algebraic and number-theoretic input such as sets of square-free (or more generally $k$-power free and $\mathcal{B}$-free) integers in number fields. Initial results for quadratic and cyclotomic number fields suggest that the fundamental invariants of orders of number fields play an important role. We expect new phenomena for non-abelian number fields. Exploration of examples will start with quadratic and cyclotomic number fields and then move to other fields. In a broader perspective, our ambition is to replace the ambient space $\mathbb{R}^n$ by a locally compact abelian group.