We will present a refinement, developed in collaboration with Bjorklund, of Furstenberg's ergodic-theoretic approach tailored to addressing the problem of identifying  'twisted infinite patterns’ in positive-density subsets of the integer lattice. These patterns correspond to an infinite structure within the 'sum-products' formed by such sets. The talk is based on joint work with Bjorklund, Bulinski, and Skinner.