Escaping Chaos in Random Multiplicative Functions

Abstract

Let $f(n)$ be a Steinhaus random multiplicative function. Let $A\subset [1,N]$ be a finite set of integers. We show that \[\frac{1}{\sqrt{|A|}} \sum_{n\in A} f(n) \xrightarrow[]{d} \mathcal{CN}(0,1)\] forces that $|A|=o(N)$. We prove that the $o(1)$ density is sharp by showing that for most sets $A$, and thus confirm the existence, with density $\ddot{I}$ such that $(1-\ddot{I}{)}^{-1}=o((\log \log N{)}^{1/2})$, we have \[ \frac{1}{\sqrt{(1-ρ) |A|}} \sum_{n\in A} f(n) \xrightarrow{d} \mathcal{CN}(0,1). \] The extra factor $\sqrt{1-\ddot{I}}$ makes a difference as long as the density $\ddot{I}>0$.

7 pages. Typos corrected