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arXiv · 2510.06161

Distribution of mixed character sums and extremal problems for Littlewood polynomials

Jonathan W. Bober·Oleksiy Klurman·Besfort Shala
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Abstract

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.

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Jonathan W. Bober, Oleksiy Klurman, Besfort Shala. 2025-10-07. Distribution of mixed character sums and extremal problems for Littlewood polynomials. https://arxiv.org/abs/2510.06161

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