Distribution of mixed character sums and extremal problems for Littlewood polynomials
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arXiv
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| Autori principali: | , , |
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| Natura: | Preprint |
| Pubblicazione: |
2025
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| _version_ | 1866914397111189504 |
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| author | Bober, Jonathan W. Klurman, Oleksiy Shala, Besfort |
| author_facet | Bober, Jonathan W. Klurman, Oleksiy Shala, Besfort |
| contents | 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 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 $α=1/4,$ proving a conjecture of Günther and Schmidt. An important ingredient in our work is a general way of dealing with "log-integrability" problems. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2510_06161 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Distribution of mixed character sums and extremal problems for Littlewood polynomials Bober, Jonathan W. Klurman, Oleksiy Shala, Besfort Number Theory Classical Analysis and ODEs Complex Variables Probability 11C08, 11L40, 11R06, 26D10, 42A05, 60G50 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 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 $α=1/4,$ proving a conjecture of Günther and Schmidt. An important ingredient in our work is a general way of dealing with "log-integrability" problems. |
| title | Distribution of mixed character sums and extremal problems for Littlewood polynomials |
| topic | Number Theory Classical Analysis and ODEs Complex Variables Probability 11C08, 11L40, 11R06, 26D10, 42A05, 60G50 |
| url | https://arxiv.org/abs/2510.06161 |