Moments of polynomials with random multiplicative coefficients

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estimates (i) give proof an almost sure Salem-Zygmund type central limit theorem $P_N(\theta)$, previously obtained unpublished work Harper by different methods, (ii) asymptotically surely N)^{1/6 - \varepsilon} \max_\theta |P_N(\theta)| \exp((\log N)^{1/2+\varepsilon}), all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