论文标题
是否存在平坦的偏斜型利特伍德多项式？
Do flat skew-reciprocal Littlewood polynomials exist?
论文作者
论文摘要
$ \ { - 1,1 \} $中具有系数的多项式称为Littlewood多项式。利用rudin-shapiro多项式的特殊特性和近似理论的经典结果，例如杰克逊定理，delaValléePoussinsums，伯恩斯坦的不平等，Riesz's Lemma，差异等等。存在绝对常数$η_2>η_1> 0 $和一个序列$（p_n）$ of Littlewood polyenmials $ p_n $ of度量$ n $，以便$ qum qult {n} \ sqrt {n} \ leq | p_n（z）| \ leqη_2\ sqrt {n} \ ,, \ qquad z \ in \ mathbb {c} \ ,, \，\，\，| z | = 1,,$$ confirming a conjecture of Littlewood from 1966. Moreover, the existence of a sequence $(P_n)$ of Littlewood polynomials $P_n$ is shown in a way that in addition to the above flatness properties a certain symmetry is satisfied by the coefficients of $P_n$ making the Littlewood polynomials $P_n$ close to skew-reciprocal.
Polynomials with coefficients in $\{-1,1\}$ are called Littlewood polynomials. Using special properties of the Rudin-Shapiro polynomials and classical results in approximation theory such as Jackson's Theorem, de la Vallée Poussin sums, Bernstein's inequality, Riesz's Lemma, divided differences, etc., we give a significantly simplified proof of a recent breakthrough result by Balister, Bollobás, Morris, Sahasrabudhe, and Tiba stating that there exist absolute constants $η_2 > η_1 > 0$ and a sequence $(P_n)$ of Littlewood polynomials $P_n$ of degree $n$ such that $$η_1 \sqrt{n} \leq |P_n(z)| \leq η_2 \sqrt{n}\,, \qquad z \in \mathbb{C}\,, \, \, |z| = 1,,$$ confirming a conjecture of Littlewood from 1966. Moreover, the existence of a sequence $(P_n)$ of Littlewood polynomials $P_n$ is shown in a way that in addition to the above flatness properties a certain symmetry is satisfied by the coefficients of $P_n$ making the Littlewood polynomials $P_n$ close to skew-reciprocal.
