On the maxima of Littlewood polynomials on $[-1,1]$
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arXiv
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| Auteurs principaux: | , |
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| Format: | Preprint |
| Publié: |
2026
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| _version_ | 1866911669071904768 |
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| author | Letwin, Brayden Sawhney, Mehtaab |
| author_facet | Letwin, Brayden Sawhney, Mehtaab |
| contents | A Littlewood polynomial is a polynomial of the form \[
f_n(x)=\sum_{k=0}^n \varepsilon_k x^k \] with $\varepsilon_k\in\{-1, 1\}$. Let $(\varepsilon_k)_{k \ge 0}$ be i.i.d. Rademacher coefficients. We show that the lower envelope of $\max_{x\in[-1,1]}|f_n(x)|$ is determined by the small-ball probability of a certain Gaussian process. In particular, almost surely, \[ \liminf_{n\to\infty} \frac{\log(\max_{x\in[-1,1]}|f_n(x)|/\sqrt n)}{(\log\log n)^{1/3}}
= -\Big(\frac{3π^2}{4}\Big)^{1/3}. \] |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2604_19294 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | On the maxima of Littlewood polynomials on $[-1,1]$ Letwin, Brayden Sawhney, Mehtaab Probability Combinatorics A Littlewood polynomial is a polynomial of the form \[ f_n(x)=\sum_{k=0}^n \varepsilon_k x^k \] with $\varepsilon_k\in\{-1, 1\}$. Let $(\varepsilon_k)_{k \ge 0}$ be i.i.d. Rademacher coefficients. We show that the lower envelope of $\max_{x\in[-1,1]}|f_n(x)|$ is determined by the small-ball probability of a certain Gaussian process. In particular, almost surely, \[ \liminf_{n\to\infty} \frac{\log(\max_{x\in[-1,1]}|f_n(x)|/\sqrt n)}{(\log\log n)^{1/3}} = -\Big(\frac{3π^2}{4}\Big)^{1/3}. \] |
| title | On the maxima of Littlewood polynomials on $[-1,1]$ |
| topic | Probability Combinatorics |
| url | https://arxiv.org/abs/2604.19294 |