Special Year Informal Seminar

Following Birkhoff's proof of the Pointwise Ergodic Theorem, it has been studied whether convergence still holds along various subsequences. In 2020, Bergelson and Richter showed that under the additional assumption of unique ergodicity, pointwise convergence holds along the number theoretic sequence $\Omega(n)$, where $\Omega(n)$ denotes the number of prime factors on $n$ counted with multiplicity. In this talk, we will see that removing this assumption, a pointwise ergodic theorem does not hold along $\Omega(n)$. Time permitting, we will then study the interplay of the dynamics with certain number theoretic properties of $\Omega(n)$ to obtain further information on the asymptotic behavior of this sequence.