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 $$\frac{1}{N^2}\sum _{1\le m,n\le N}f(T^{\mathrm{\Omega }(m^2+n^2)}x)$$ 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 $\mathrm{\Omega }$ 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.