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 }\chi(n)e(n\theta), \end{equation*} for fixed $\theta\in [0,1]$ and random character $\chi \pmod q$, as well as for a fixed character $\chi$ and randomly sampled $\theta\in [0,1].$ 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 $\alpha=1/4,$ proving a conjecture of G\"unther and Schmidt. An important ingredient in our work is a general way of dealing with "log-integrability" problems.