Simplices in large sets and directional expansion in ergodic actions
Copyright policy )Simplices in large sets and directional expansion in ergodic actions
Abstract 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.
Relations between ergodic theory and number theory, 11B30, Ergodic theorems, spectral theory, Markov operators, 37A44, Dynamical systems involving one-parameter continuous families of measure-preserving transformations, 37A30, Arithmetic combinatorics; higher degree uniformity, cyclic subgroups, QA1-939, Mathematics - Combinatorics, ergodic \(\mathbb{Z}^r\)-actions, infinite arithmetic progression, Mathematics - Dynamical Systems, Mathematics
Relations between ergodic theory and number theory, 11B30, Ergodic theorems, spectral theory, Markov operators, 37A44, Dynamical systems involving one-parameter continuous families of measure-preserving transformations, 37A30, Arithmetic combinatorics; higher degree uniformity, cyclic subgroups, QA1-939, Mathematics - Combinatorics, ergodic \(\mathbb{Z}^r\)-actions, infinite arithmetic progression, Mathematics - Dynamical Systems, Mathematics
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