关于Erdős-Littlewood多项式的L^α平坦性
经典分析与常微分方程
2025-05-01 v1 动力系统
摘要
我们证明了，当α > 2为偶数时（因此对所有α ≥ 4），Erdős-Littlewood多项式不是L^α-平坦的。这为Littlewood提出的一个老问题提供了部分解答。作为推论，我们得到了Erdős-Newman猜想在系数为±1的多项式上的肯定回答；即，Erdős-Littlewood多项式类中不存在超平坦多项式序列。我们的证明简短而简洁。它依赖于Dirichlet核的L^p范数的经典引理、Marcinkiewicz-Zygmund插值不等式以及A. Bonami和S. Révész提出的p-集中定理。
引用
@article{arxiv.2504.21499,
title = {On $L^\alpha$-flatness of Erd\H{o}s-Littlewood's polynomials},
author = {el Houcein el Abdalaoui},
journal= {arXiv preprint arXiv:2504.21499},
year = {2025}
}
备注
10 pages, 27 references and three quotations: <<Begin at the beginning, the King said gravely, "and go on till you come to the end: then stop,''>> by Lewis Carroll, <<Those who know do not speak; those who speak do not know,>> by Laozi and <<Everyone writes, nobody reads,>> by Erd\H{o}s (which should be attributed to Fej\'er according to Erd\H{o}s)