Conjectured optimal convergence criterion for Hardy-field functions

Let $\mathcal{H}$ be the Hardy field under consideration, and let $a_1,\ldots,a_k\in\mathcal{H}$. Write $e(u)=\exp(2\pi i u)$, and let $(X,\mu,T)$ be a measure-preserving system with $f_1,\ldots,f_k\in L^{\infty}(\mu)$. The functions $a_1,\ldots,a_k$ are assumed to satisfy that, for every $t_1,\ldots,t_k\in[0,1)$, the averages

$$\frac{1}{N}\sum_{n=1}^{N}e\left(t_1\lfloor a_1(n)\rfloor+\cdots+t_k\lfloor a_k(n)\rfloor\right)$$

converge.

Conjectured optimal convergence criterion. Under this assumption, the averages

$$\frac{1}{N}\sum_{n=1}^{N}T^{\lfloor a_1(n)\rfloor}f_1\cdots T^{\lfloor a_k(n)\rfloor}f_k$$

converge in $L^2(\mu)$ for every measure-preserving system $(X,\mu,T)$. If $(X,\mu,T)$ is a nilsystem, then they converge pointwise.

The condition is conjectured to be optimal because it is precisely the condition that the functions are good for convergence when the system is a rotation on a torus. The preceding theorem establishes $L^2$ convergence under stronger hypotheses, while the pointwise nilsystem assertion gives a stronger conclusion in that special class; the general criterion remains unresolved in the source.