Bounds for exponential sums with random multiplicative coefficients

For $f$ a Rademacher or Steinhaus random multiplicative function, we prove that $$\max_{\theta \in [0,1]} \frac{1}{\sqrt{N}} \Bigl| \sum_{n \leq N} f(n) \mathrm{e} (n \theta) \Bigr| \gg \sqrt{\log N} ,$$ asymptotically almost surely as $N \rightarrow \infty$. Furthermore, for $f$ a Steinhaus random multiplicative function, and any $\varepsilon > 0$, we prove the partial upper bound result $$\max_{\theta \in [0,1]} \frac{1}{\sqrt{N}} \Bigl| \sum_{\substack{n \leq N \\ P(n) \geq N^{0.8}}} f(n) \mathrm{e} (n \theta) \Bigr| \ll {(\log N)}^{7/4 + \varepsilon},$$ asymptotically almost surely as $N \rightarrow \infty$, where $P(n)$ denotes the largest prime factor of $n$.