Bounds for exponential sums with random multiplicative coefficients

For $f$ a Rademacher or Steinhaus random multiplicative function, we prove that $$\underset{\theta \in [0,1]}{\max }\frac{1}{\sqrt{N}}\mid \sum _{n\le N}f(n)\mathrm{e}(n\theta )\mid \gg \sqrt{\log N},$$ asymptotically almost surely as $N\to \mathrm{\infty }$. Furthermore, for $f$ a Steinhaus random multiplicative function, and any $\epsilon >0$, we prove the partial upper bound result $$\underset{\theta \in [0,1]}{\max }\frac{1}{\sqrt{N}}\mid \sum _{\begin{array}{c}n\le N\\ P(n)\ge N^{0.8}\end{array}}f(n)\mathrm{e}(n\theta )\mid \ll {(\log N)}^{7\mathrm{/}4+\epsilon },$$ asymptotically almost surely as $N\to \mathrm{\infty }$, where $P(n)$ denotes the largest prime factor of $n$.