  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.
