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 exhibit the strong sweeping‐out property, which answers a question in the affirmative posed by Le, Moreira, Sun, and Donoso. On the other hand, using number‐theoretic methods, we establish the pointwise convergence of averages along the 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.