Escaping Chaos in Random Multiplicative Functions

Let $f(n)$ be a Steinhaus random multiplicative function. Let $A\subset [1,N]$ be a finite set of integers. We show that $$\frac{1}{\sqrt{\mathrm{\mid }A\mathrm{\mid }}}\sum _{n\in A}f(n)\underset{}{\overset{d}{\to }}\mathcal{C}\mathcal{N}(0,1)$$ forces that $\mathrm{\mid }A\mathrm{\mid }=o(N)$. We prove that the $o(1)$ density is sharp by showing that for most sets $A$, and thus confirm the existence, with density $\rho$ such that $(1-\rho {)}^{-1}=o((\log \log N{)}^{1\mathrm{/}2})$, we have $$\frac{1}{\sqrt{(1-\rho )\mathrm{\mid }A\mathrm{\mid }}}\sum _{n\in A}f(n)\stackrel{d}{\to }\mathcal{C}\mathcal{N}(0,1)\mathrm{.}$$ The extra factor $\sqrt{1-\rho }$ makes a difference as long as the density $\rho >0$.