Some ergodic theorems over $k$-full numbers

In 2022, Bergelson and Richter established a new dynamical generalization of the prime number theorem. Later, Loyd showed a disjoint form with the Erd\H{o}s-Kac theorem. Recently, the author and his coauthors proved some ergodic theorems over squarefree numbers related to these results. In this paper, building on the previous work, we will derive the analogues of Bergelson-Richter's theorem, Erd\H{o}s-Kac theorem and Loyd's theorem over $k$-full numbers for any integer $k\geq2$.