Result: Simplices in large sets and directional expansion in ergodic actions
Title:
Simplices in large sets and directional expansion in ergodic actions
Authors:
Source:
Forum of Mathematics, Sigma, Vol 12 (2024)
Publication Status:
Preprint
Publisher Information:
Cambridge University Press (CUP), 2024.
Publication Year:
2024
Subject Terms:
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
Document Type:
Academic journal
Article
File Description:
application/xml
Language:
English
ISSN:
2050-5094
DOI:
10.1017/fms.2024.125
Access URL:
Rights:
CC BY
Accession Number:
edsair.doi.dedup.....d006c0a973a94e19b339eadef496ea07
Database:
OpenAIRE
Further Information
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.