This talk will provide some motivation for and an introduction to recent joint work with Caroline Terry.

Our previous joint work had identified model-theoretic stability as a sufficient condition for the existence of strong arithmetic regularity decompositions in finite abelian groups. Ultimately based on Tim Gowers’s groundbreaking work on Szemerédi’s theorem in the late 90s, *higher-order* arithmetic regularity decompositions are nowadays an essential part of the arithmetic combinatorialist’s toolkit.

We define a natural higher-order generalisation of stability and prove that it implies the existence of particularly efficient higher-order arithmetic regularity decompositions in the setting of elementary p-groups. If time permits, I will briefly outline some analogous results we obtain in the hypergraph setting.