Mean Li-Yorke chaos along some good sequences

If a topological dynamical system $(X,T)$ has positive topological entropy, then it is multivariant mean Li-Yorke chaotic along a sequence $\{a_k{\}}_{k=1}^{\mathrm{\infty }}$ of positive integers which is “good” for pointwise ergodic convergence with a mild condition; more specifically, there exists a Cantor subset $K$ of $X$ such that for every $n\ge 2$ and pairwise distinct points $x_1,x_2,\dots ,x_n$ in $K$ we have $$\underset{N\to \mathrm{\infty }}{\mathrm{lim\hspace{0.17em}inf}}\frac{1}{N}\sum _{k=1}^N\underset{1\le i<j\le n}{\max }d(T^{a_k}{x}_i,T^{a_k}{x}_j)=0$$ and $$\underset{N\to \mathrm{\infty }}{\mathrm{lim\hspace{0.17em}sup}}\frac{1}{N}\sum _{k=1}^N\underset{1\le i<j\le n}{\min }d(T^{a_k}{x}_i,T^{a_k}{x}_j)>0.$$ Examples are given for the classic sequences of primes and generalized polynomials.