Abstract: Given a simplicial projective toric variety with at most one singular point, we study the asymptotic structure of the sumsets arising from its parametrization. We introduce a notion of sumsets regularity and relate it to the Castelnuovo–Mumford regularity of the variety, translating an algebraic problem into a combinatorial one. We then obtain new upper bounds on the sumset regularity, and hence on the Castelnuovo–Mumford regularity of these varieties. This talk is based on joint work with Ignacio García-Marco and Philippe Gimenez.