$Δ$-weakly mixing subsets along a collection of sequences of integers

Abstract: In this paper, we propose a mild condition, named Condition $(<strong>)$, for collections of sequence of integers and show that for any measure preserving system the Pinsker $\sigma$-algebra is a characteristic $\sigma$-algebra for the averages along a collection satisfying Condition $(</strong>)$. We introduce the notion of $\Delta$-weakly mixing subsets along a collection of sequences of integers and show that positive topological entropy implies the existence of $\Delta$-weakly mixing subsets along a collection of "good" sequences. As a consequence, we show that positive topological entropy implies multi-variant Li-Yorke chaos along polynomial times of the shift prime numbers.