Calder\'on-type inequalities for affine frames

We prove sharp upper and lower bounds for generalized Calder'on’s sums associated to frames on LCA groups generated by affine actions of cocompact subgroup translations and general measurable families of automorphisms. The proof makes use of techniques of analysis on metric spaces, and relies on a counting estimate of lattice points inside metric balls. We will deduce as special cases Calder'on-type inequalities for families of expanding automorphisms as well as for LCA-Gabor systems.