Distribution of mixed character sums and extremal problems for Littlewood polynomials

TL;DR

This work studies the distribution of mixed character sums $S(\chi,x,\theta)$ in two regimes (random $\chi$ with fixed $\theta$ and fixed $\chi$ with random $\theta$) by introducing normalized random processes $F_{k,\chi,\alpha,\beta}(t)$ and $F_{\alpha,\beta}(t)$, along with a quadratic-character analog $G_{q,\alpha,\beta}$ and its limit $G_{\alpha,\beta}$. The authors develop a proof framework built on a twisted Poisson summation formula, truncation, moment convergence via Prokhorov’s method, and prime-removal lemmas, augmented by a novel log-integrability technique to control small-value contributions. These distributional results yield two major applications: constructing Littlewood polynomials with a normalized Mahler measure exceeding $0.954$, thus breaking a record in the Mahler problem, and confirming that the extremal $L_{2k}$ norms of Turyn/Fekete-type polynomials occur at a shift $\alpha=1/4$, validating the Günther–Schmidt conjecture. The paper also develops a general log-integrability principle that has potential beyond these problems, enabling robust handling of logarithmic integrals in incomplete exponential sums and random processes. Overall, the results provide a new probabilistic framework for incomplete exponential sums and concrete extremal consequences for Littlewood polynomials with implications for analytic number theory and polynomial extremality.

Abstract

We prove distributional results for mixed character sums \begin{equation*} \sum_{n\le x }χ(n)e(nθ), \end{equation*} for fixed $θ\in [0,1]$ and random character $χ\pmod q$, as well as for a fixed character $χ$ and randomly sampled $θ\in [0,1].$ We present various applications of our results. For example, we construct Littlewood polynomials with large Mahler measure, thus establishing a new record in the Mahler problem (1963). We also show that $L_{2k}$ norms of well-known Turyn polynomials are asymptotically minimized at the shift $α=1/4,$ proving a conjecture of Günther and Schmidt. An important ingredient in our work is a general way of dealing with "log-integrability" problems.

Theorems & Definitions (46)

• Theorem 1
• Corollary 1
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• Corollary 2
• Theorem 3
• Theorem 4
• Proposition 1
• Proposition 2
• Lemma 1
• proof