A subset generalization of the Erdős-Kac theorem over number fields with applications

Let $\omega (n)$ denote the number of distinct prime factors of a natural number $n$. In 1940, Erdős and Kac established that $\omega (n)$ obeys the Gaussian distribution over natural numbers. In 2004, the third author generalized their theorem to all abelian monoids. In this work, we extend the work of the third author to any subset of the set of ideals of a number field satisfying some additional conditions. Finally, we apply this theorem to prove the Erdős-Kac theorem over $h$-free and over $h$-full ideals of the number field.