Do sets of big returns contain sets of returns?

Question

We say a subset $E$ of $\mathbb{N}$ is a set of returns if there is some measure preserving system $(X,\mathcal{B},\mu,T)$ and some $A\in\mathcal{B}$ with $\mu(A)>0$ such that $E=\{n\in\mathbb{N};\mu(A\cap T^{-n}A)>0\}$.

We say a subset $E$ of $\mathbb{N}$ is a set of big returns if there is some measure preserving system $(X,\mathcal{B},\mu,T)$ and some $A\in\mathcal{B},\varepsilon>0$ with $\varepsilon<\mu(A)^2$ such that $E=\{n\in\mathbb{N};\mu(A\cap T^{-n}A)>\varepsilon\}$.

Clearly, every set of returns contains a set of big returns. Also, some properties I know about sets of returns are also satisfied by sets of big returns (e.g. being syndetic).

My question is, does every set of big returns contain some set of returns?

Answer 1

No, a counterexample is due to Alan Forrest:

Forrest, A. H., The construction of a set of recurrence which is not a set of strong recurrence, Isr. J. Math. 76, No. 1-2, 215-228 (1991). ZBL0773.28014.

McCutcheon has a simpler construction along the same lines:

McCutcheon, Randall, Three results in recurrence, Petersen, Karl E. (ed.) et al., Ergodic theory and its connections with harmonic analysis. Proceedings of the 1993 Alexandria conference, Alexandria, Egypt, May 24-28, 1993. Cambridge: Cambridge University Press. Lond. Math. Soc. Lect. Note Ser. 205, 349-358 (1995). ZBL0867.28010.

I constructed examples where no set of big returns contains a translate of a set of returns: Corollary 2.5 in

Griesmer, John T., Recurrence, rigidity, and popular differences, Ergodic Theory Dyn. Syst. 39, No. 5, 1299-1316 (2019). ZBL1512.37004.

Ackelsberg generalized this to countable abelian groups:

Ackelsberg, Ethan M., Rigidity, weak mixing, and recurrence in abelian groups, Discrete Contin. Dyn. Syst. 42, No. 4, 1669-1705 (2022). ZBL1498.37004.