Large fluctuations of sums of a random multiplicative function

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.