Quantitative multiple recurrence for two and three transformations.

In this talk I will provide some counter-examples for quantitative multiple recurrence problems for systems with more than one transformation. For instance, I will show that there exists an ergodic system $(X,\mathcal{X},\mu,T_1,T_2)$ with two commuting transformations such that for every $\ell < 4$ there exists $A\in \mathcal{X}$ such that $\mu(A\cap T_1^n A\cap T_2^n A) < \mu(A)^{\ell}$ for every $n \in \mathbb{N}$. The construction of such a system is based on the study of ``big’’ subsets of $\mathbb{N}^2$ and $\mathbb{N}^3$ satisfying combinatorial properties.

This a joint work with Wenbo Sun.