A comment on the number of $k$-th powers inside arithmetic progressions

In \cite{BD} Bourgain and Demeter found sharp upper bounds for the number of $k$-th powers inside arbitrary arithmetic progressions whose step has $O(1)$ many divisors.

We make the easy observation that the same arguments are still valid if the step does not grow too rapidly in relation to the length of the progression.

Furthermore, we give sharp bounds for the number of $k$-th powers among the first $N$ terms for $N$ large enough.

Both results should be known.

Nevertheless, we add to the literature.