On the number of distinct prime factors of $nj+a^{h}k$

Let $\omega (n)$ denote the number of distinct prime factors of $n$. Then for any given $K\ge 2$, small $ϵ>0$ and sufficiently large (only depending on $K$ and $ϵ$) $x$, there exist at least $x^{1-ϵ}$ integers $n\in [x,(1+K^{-1})x]$ such that $\omega(nj\pm a^hk)\geq(\log\log\log x)^{{1/3}-\epsilon}$forall$2\leq a\leq K$,$1\leq j,k\leq K$and$0\leq h\leq K\log x$.