Simplices in large sets and directional expansion in ergodic actions

Journal article, 2024

In this paper, we study ergodic $\mathbb {Z}<^>r$ -actions and investigate expansion properties along cyclic subgroups. We show that under some spectral conditions, there are always directions which expand significantly a given measurable set with positive measure. Among other things, we use this result to prove that the set of volumes of all r-simplices with vertices in a set with positive upper density must contain an infinite arithmetic progression, thus showing a discrete density analogue of a classical result by Graham.