It is well known that the density of $x^\alpha$ smooth integers up to $x$ has positive measure.
Now, one can ask about the difference on these $x^\alpha$ smooth integers, and hence these measure is asymptotically equal, one can easily come up with the fact that
For some integer $c$, if $y, y+c$ is both $x^\alpha$ smooth, let $y$ be "good". If $x$ is large enough, number of good integers upto $x$ has constant positive measure.
Now, one can ask if this holds for all $c$.
Which is my question.
Thanks.