Distribution of mixed character sums and extremal problems for Littlewood polynomials

We prove distributional results for mixed character sums \begin{equation*} \sum_{n\le x }χ(n)e(nθ), \end{equation*} for fixed $\theta \in [0,1]$ and random character $\chi (\mathrm{m}\mathrm{o}\mathrm{d}q)$, as well as for a fixed character $\chi$ and randomly sampled $\theta \in [0,1]\mathrm{.}$ We present various applications of our results. For example, we construct Littlewood polynomials with large Mahler measure and $L_1$ norm, thus establishing new records in the Mahler and Newman problems. We also show that $L_{2k}$ norms of well-known Turyn polynomials are asymptotically minimized at the shift $\alpha =1\mathrm{/}4,$ proving a conjecture of Günther and Schmidt. An important ingredient in our work is a general way of dealing with “log-integrability” problems.