IP-multiple recurrence for k commuting anti-actions yields a left IP* set

Establish that for every countable, discrete, amenable group G, every Lebesgue probability space (X, A, μ), every k ∈ N, and every collection of k commuting, measure-preserving anti-actions T^{(1)}, …, T^{(k)} of G, there exists λ = λ(A, T_g^{(1)}, …, T_g^{(k)}) > 0 such that the set { g ∈ G : μ(A ∩ (T_g^{(1)})^{-1}A ∩ … ∩ (T_g^{(1)}…T_g^{(k)})^{-1}A) > λ } is left IP*.

Background

The paper recalls a result of Bergelson and McCutcheon that establishes a central* (C*)-type multiple recurrence in noncommutative settings and indicates their conjecture that this can be strengthened to an IP* conclusion and extended to any number k of commuting anti-actions. The present work proves the k=1 case (Theorem 6.2), upgrading the C* conclusion to IP*, but leaves the general k≥2 case open.

This conjecture, if resolved, would imply stronger combinatorial van der Waerden-type conclusions for amenable groups in higher dimensions, connecting ergodic-theoretic multiple recurrence to IP*-largeness in the noncommutative setting.

References

In , V. Bergelson and R. McCutcheon conjectured that \cref{Bergelson_McCutcheon_density} also holds for $k$ commuting and measure preserving anti-actions, for $k \in \mathbb{N}$ arbitrary, and this would imply that also \cref{Bergelson_McCutcheon_combinatorial} holds in $Gm$ for $m \in \mathbb{N}$ arbitrary. Moreover, they conjectured that $\mathcal{C}{\ast}$ conclusion can be upgraded to IP${\ast}$ conclusion. More precisely they have conjectured the following: Let $(X,\mathcal{A},\mu )$ be a Lebesgue probability space and $T{(1)},\ldots, T{(k)}$ be commuting and measure preserving anti-actions. For any $A \in \mathcal{A}$ with $\mu(A) >0$ there exists $\lambda=\lambda(A,T_g{(1)},\ldots, T_g{(k)}) >0$ such that \begin{align*} { g \in G: \mu(A \cap (T_g{(1)}){-1}A \cap \ldots \cap (T_g{(1)} \ldots T_g{(k)}){-1}A) > \lambda } \end{align*} is left IP${\ast}$.

— Van der Waerden type theorem for amenable groups and FC-groups  (2411.15987 - Parini, 2024) in Introduction, Conjecture 1