A multiplicative action on a probability space $(X,\mathcal{A},\mu)$ is a sequence of measure-preserving transformations $S = (S_n)_{n\in\mathbb{N}}$, satisfying $S_{nm} = S_n \circ S_m$, for all $n,m \in \mathbb{N}$, and it is called \emph{finitely generated}
if the set $\{S_p : p \in \mathbb{P}\}$ is finite. In joint ongoing work with A. Koutsogiannis and K. Tsinas, we show that for any commuting finitely generated multiplicative actions $S,\tilde{S}$ on a probability space $(X,\mathcal{A},\mu)$,and for any $F,G \in L^2(\mu)$, the limit
\[
\lim_{N\to\infty}\ \frac{1}{N}\sum_{n=1}^{N} S_nF\cdot \tilde{S}_n G
\]
exists in $L^2(\mu)$. This was posed as a problem in a paper of Frantzikinakis.
Bryna Kra
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