Pointwise Ergodic Averages Along the Omega Function in Number Fields

We show the failure of the pointwise convergence of averages along the Omega function in a number field. As a consequence, we show, for instance, that the averages $$\frac{1}{N^2}\sum_{1\leq m,n \leq N} f(T^{\Omega(m^2+n^2)}x)$$ do not converge pointwise in ergodic systems, addressing 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.