Pointwise ergodic averages along the Omega function in number fields

We study a general criterion for guaranteeing that an ergodic average exhibits the strong sweeping-out property. This result implies, in particular, the failure of pointwise convergence of these averages. Our result applies to averages along the Omega function in number fields, generalizing a result of Loyd. We also show that the averages \[\frac{1}{N^2}\sum_{1\leq m,n \leq N}f(T^{\Omega(m^2+n^2)}x)\] exhibit the strong sweeping-out property, which answers affirmatively a question posed by Le, Moreira, Sun, and the second author.
On the other hand, using number-theoretic methods, we establish the pointwise convergence of averages along the $\Omega$ function defined on the ideals of a number field in uniquely ergodic systems. Using this dynamical framework, we also derive several natural number-theoretic consequences of independent interest.