Central limit theorems for random multiplicative functions

A Steinhaus random multiplicative function $f$ is a completely multiplicative function obtained by setting its values on primes $f(p)$ to be independent random variables distributed uniformly on the unit circle. Recent work of Harper shows that ${\sum }_{n\le N}f(n)$ exhibits ``more than square-root cancellation," and in particular $\frac{1}{\sqrt{N}}{\sum }_{n\le N}f(n)$ does not have a (complex) Gaussian distribution. This paper studies ${\sum }_{n\in \mathcal{A}}f(n)$, where $\mathcal{A}$ is a subset of the integers in $[1,N]$, and produces several new examples of sets $\mathcal{A}$ where a central limit theorem can be established. We also consider more general sums such as ${\sum }_{n\le N}f(n)e^{2\pi in\theta }$, where we show that a central limit theorem holds for any irrational $\theta$ that does not have extremely good Diophantine approximations.