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

Abstract

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

This is a very preliminary draft, and maybe contains some mistakes