Higher-order correlations for group actions

TL;DR

This survey explores higher-order correlations in group actions and dynamical systems, providing bounds, applications in lattice point counting, approximate configurations, and CLT validity for multi-parameter actions.

Contribution

It offers a comprehensive overview and a self-contained proof of bounds for higher-order correlations in simple Lie group actions, with diverse applications.

Findings

Quantitative bounds for higher-order correlations.

Asymptotic formulas for lattice point counting.

Validation of the Central Limit Theorem for multi-parameter actions.

Abstract

This survey paper discusses behaviour of higher-order correlations for one-parameter dynamical systems and more generally for dynamical systems arising from group actions. In particular, we present a self-contained proof of quantitative bounds for higher-order correlations of actions of simple Lie groups. We also outline several applications of our analysis of correlations that include asymptotic formulas for counting lattice points, the existence of approximate configurations in lattice subgroups, and validity of the Central Limit Theorem for multi-parameter group actions.