Aperiodic multiplicative orthogonality conjecture for homogeneous ergodic averages

Let $\bm \nu$ be an aperiodic bounded multiplicative function and let $l\geq 2$ be a positive integer. Let $T_1,\ldots,T_l$ be commuting measure-preserving transformations acting on the same probability space $(X,\mathcal{A},\mu)$, and let $f_1,\ldots,f_l\in L^\infty(X,\mu)$. Aperiodic multiplicative orthogonality conjecture. For almost every $x\in X$,

$$\frac{1}{N}\sum_{n=1}^{N}\bm \nu(n)\prod_{j=1}^{l}f_j(T_j^n x)\longrightarrow 0\quad\text{as }N\to+\infty.$$

This conjecture predicts almost-sure cancellation of homogeneous multilinear ergodic averages weighted by any aperiodic bounded multiplicative function, including the Möbius and Liouville functions. It extends the corresponding convergence results established in the paper, while the general case remains open.