Better than square-root cancellation for random multiplicative functions

We investigate when the better than square-root cancellation phenomenon exists for ${\sum }_{n\le N}a(n)f(n)$, where $a(n)\in \mathbb{C}$ and $f(n)$ is a random multiplicative function. We focus on the case where $a(n)$ is the indicator function of $R$ rough numbers. We prove that $\log \log R\asymp (\log \log x{)}^{\frac{1}{2}}$ is the threshold for the better than square-root cancellation phenomenon to disappear.