Large fluctuations of sums of a random multiplicative function

TL;DR

This paper develops a framework to analyze large fluctuations of sums of random multiplicative functions over various subsets, extending central limit theorem results and establishing almost sure bounds and limsup behavior.

Contribution

It extends the CLT framework for random multiplicative functions to new subsets like short intervals and polynomial values, providing almost sure fluctuation bounds.

Findings

Almost sure limsup of sums over short intervals is positive.

Almost sure limsup of sums over polynomial values exceeds a constant.

Established upper bounds matching the law of iterated logarithm.

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.