Central limit theorems for random multiplicative functions

In joint work with Kannan Soundararajan, we consider the behavior of random multiplicative functions when summed over subsets of the integers in [1, N], and give several examples of sets where such sums satisfy a central limit theorem. In contrast, as we know from the work of Harper, the partial sums over all integers in [1, N] do not satisfy a central limit theorem. The key quantity involved in the study is the "multiplicative energy" of a set, which is an object of interest to combinatorialists.