Ergodic averages for commutative transformations along return times

Abstract: In this paper, we extend recent results on the convergence of ergodic averages along sequences generated by return times to shrinking targets in rapidly mixing systems, partially answering questions posed by the first author, Maass and the third author in [6]. In particular, for a fixed parameter $a\in (0,1)$ and for generic $y\in [0,1]$, we establish both $L^2$ and pointwise convergence for single averages and multiple averages for commuting transformations along the sequences $(a_n(y))_{n\in \mathbb{N}}$, obtained by arranging the set $$\Big{n\in\mathbb{N}: 0<2^ny \mod{1}<n^{-a} \Big}$$ in an increasing order. We also obtain new results for semi-random ergodic averages along sequences of similar type.