# Quantitative multiple recurrence for two and three transformations

## TL;DR

This paper constructs counterexamples in ergodic theory showing that certain multiple recurrence properties do not hold universally for systems with multiple commuting transformations, especially in higher complexity settings.

## Contribution

It provides explicit counterexamples demonstrating the failure of quantitative multiple recurrence for systems with two or three transformations and for systems generating 2-step nilpotent groups.

## Key findings

- Existence of ergodic systems with two transformations where recurrence bounds fail for all exponents less than 4.
- Existence of ergodic systems with three transformations where recurrence bounds fail for all positive exponents.
- Existence of systems with two transformations generating a 2-step nilpotent group where recurrence bounds fail for all positive exponents.

## Abstract

We provide various counter examples for quantitative multiple recurrence problems for systems with more than one transformation. We show that $\bullet$ There exists an ergodic system $(X,\mathcal{X},\mu,T_1,T_2)$ with two commuting transformations such that for every $0<\ell< 4$, there exists $A\in\mathcal{X}$ such that $$\mu(A\cap T_{1}^{-n}A\cap T_{2}^{-n}A)<\mu(A)^{\ell} \text{ for every } n\neq 0;$$ $\bullet$ There exists an ergodic system $(X,\mathcal{X},\mu,T_1,T_2, T_{3})$ with three commuting transformations such that for every $\ell>0$, there exists $A\in\mathcal{X}$ such that $$\mu(A\cap T_{1}^{-n}A\cap T_{2}^{-n}A\cap T_{3}^{-n}A)<\mu(A)^{\ell} \text{ for every } n\neq 0;$$ $\bullet$ There exists an ergodic system $(X,\mathcal{X},\mu,T_1,T_2)$ with two transformations generating a 2-step nilpotent group such that for every $\ell>0$, there exists $A\in\mathcal{X}$ such that $$\mu(A\cap T_{1}^{-n}A\cap T_{2}^{-n}A)<\mu(A)^{\ell} \text{ for every } n\neq 0.$$