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.
