Moments of polynomials with random multiplicative coefficients

TL;DR

This paper studies the moments of random polynomials with multiplicative coefficients, showing Gaussian behavior for certain moments and establishing almost sure bounds on their maximum, with implications for central limit theorems.

Contribution

It proves Gaussian moment convergence for polynomials with multiplicative coefficients and derives almost sure bounds on their maxima, extending previous results with new methods.

Findings

Moments tend to Gaussian moments for k up to (log N / log log N)^{1/3}.

Almost sure bounds on the maximum of P_N(θ) are established.

An almost sure CLT for P_N(θ) is proved, confirming previous unpublished results.

Abstract

For $X(n)$ a Rademacher or Steinhaus random multiplicative function, we consider the random polynomials $$ P_N(\theta) = \frac1{\sqrt{N}} \sum_{n\leq N} X(n) e(n\theta), $$ and show that the $2k$-th moments on the unit circle $$ \int_0^1 \big| P_N(\theta) \big|^{2k}\, d\theta $$ tend to Gaussian moments in the sense of mean-square convergence, uniformly for $k \ll (\log N / \log \log N)^{1/3}$, but that in contrast to the case of i.i.d. coefficients, this behavior does not persist for $k$ much larger. We use these estimates to (i) give a proof of an almost sure Salem-Zygmund type central limit theorem for $P_N(\theta)$, previously obtained in unpublished work of Harper by different methods, and (ii) show that asymptotically almost surely $$ (\log N)^{1/6 - \varepsilon} \ll \max_\theta |P_N(\theta)| \ll \exp((\log N)^{1/2+\varepsilon}), $$ for all $\varepsilon > 0$.