Large fluctuations of sums of a random multiplicative function

Abstract

Let $f$ be a Rademacher or Steinhaus random multiplicative function. For various arithmetically interesting subsets $\mathcal A\subseteq [1, N]\cap\mathbb N$ such that the distribution of $\sum_{n\in \mathcal A} f(n)$ is approximately Gaussian, we develop a general framework to understand the large fluctuations of the sum. This extends the general central limit theorem framework of Soundararajan and Xu. In the case when $\mathcal A = (N-H, N]$ is a short interval with admissible $H=H(N)$, we show that almost surely \begin{equation*} \limsup_{N\to\infty} \frac{\big\lvert\sum_{N-H<n\leq N} f(n)\big\rvert}{\sqrt{H\log \frac{N}H{}}}>0. \end{equation*} When $\mathcal A$ is the set of values of an admissible polynomial $P\in\mathbb Z[x]$, we extend work of Klurman, Shkredov, and Xu, as well as Chinis and the author, showing that almost surely \begin{equation*} \limsup_{N\to\infty} \frac{\big\lvert\sum_{n\leq N} f(P(n))\big\rvert}{\sqrt{N \log\log N}}>0, \end{equation*} even when $P$ is a product of linear factors over $\mathbb Q$. In this case, we also establish the corresponding almost sure upper bound, matching the law of iterated logarithm. An important ingredient in our work is bounding the Kantorovich--Wasserstein distance by means of a quantitative martingale central limit theorem.