论文标题
关于rudin-shapiro的模量的振荡，其范围中间
On the oscillations of the modulus of Rudin-Shapiro polynomials around the middle of their ranges
论文作者
论文摘要
让$ r_k（t）：= | p_k（e^{it}）|^2 $或$ r_k（t）：= | q_k（e^{it}）|^2 $，其中$ p_k $和$ q_k $是通常的rudin-shapiro poludin-shapiro poludin-shapiro polynomials frem $ n-1 $ n = 2^k $。在此期间，$ r_k $的图表表明，在此期间，以浓密的方式以$ r_k（t）-n $的数量为单位。令$ n（i，r_k-n）$表示间隔$ i中$ r_k-n $的零数：= [α，β] \ subset [0,2π] $。改进早期结果仅针对$ i：= [0,2π] $，在本文中，我们表明$$ \ frac {n | i |} {8π} {8π} - \ frac {2}π（2n \ log n）^{1/2} {1/2} {1/2} - 1 \ leq n（i，r_k -n） \ frac {8}π（2n \ log n）^{1/2} \ ,, \ qQuad k \ geq 2 \ ,, $ i：$ i：= [α，β] \ subset [0,2π] $，$ | i | =β-α$表示间隔$ i $的长度。
Let either $R_k(t) := |P_k(e^{it})|^2$ or $R_k(t) := |Q_k(e^{it})|^2$, where $P_k$ and $Q_k$ are the usual Rudin-Shapiro polynomials of degree $n-1$ with $n=2^k$. The graphs of $R_k$ on the period suggest many zeros of $R_k(t)-n$ in a dense fashion on the period. Let $N(I,R_k-n)$ denote the number of zeros of $R_k-n$ in an interval $I := [α,β] \subset [0,2π]$. Improving earlier results stated only for $I := [0,2π]$, in this paper we show that $$\frac{n|I|}{8π} - \frac{2}π (2n\log n)^{1/2} - 1 \leq N(I,R_k-n) \leq \frac{n|I|}π + \frac{8}π(2n\log n)^{1/2}\,,\qquad k \geq 2\,,$$ for every $I := [α,β] \subset [0,2π]$, where $|I| = β-α$ denotes the length of the interval $I$.
