Roth's Theorem in Super Smooth Numbers

We say that the set of $y$-smooth numbers $\mathcal{S}(N,y)$ up to $N$ is super smooth if $y=\log^KN$ for a large fixed constant $K$.

We show that the Roth's theorem on arithmetic progressions is true in super smooth numbers case.

This extends the result of Harper where he showed the statement is true under a weaker hypothesis.