Direct and inverse results for popular differences in trees of positive dimension

We establish analogues for trees of results relating the density of a set ${E \subset \mathbb {N}}$ , the density of its set of popular differences and the structure of E. To obtain our results, we formalize a correspondence principle of Furstenberg and Weiss which relates combinatorial data on a tree to the dynamics of a Markov process. Our main tools are Kneser-type inverse theorems for sets of return times in measure-preserving systems. In the ergodic setting, we use a recent result of the first author with Bjorklund and Shkredov and a stability-type extension (proved jointly with Shkredov); we also prove a new result for non-ergodic systems.