关于单位圆上Rudin-Shapiro多项式模的振动性
On the oscillation of the modulus of the Rudin-Shapiro polynomials on the unit circle
摘要：
在信号处理中，Rudin-Shapiro多项式具有良好的自相关特性，并且在单位圆上的值很小。具有低自相关系数的二进制序列在雷达、声纳和通信系统中具有重要意义。本文研究了单位圆上Rudin-Shapiro多项式模的振动性。我们还证明了次数为$n-1$且含$n := 2^k$的Rudin-Shapiro多项式$P_k$和$Q_k$在单位圆上有$o(n)$零点.这应该与B的结果进行比较。Conrey，A.格兰维尔湾Poonen和K. Soundararajan指出，对于奇素数$p$的费凯特多项式$f_p$的程度$p-1$有渐近$κ_0 p$零上的单位圆，其中$0.500813>κ_0>0.500668$。我们的方法主要基于最近由B.罗杰斯我们还证明了存在绝对常数$c_1 > 0$和$c_2 > 0$，使得$n-1 = 2^k-1$次的$k$次Rudin-Shapiro多项式$P_k$和$Q_k$在环$$\left \{z \in {\Bbb C}: 1 - \frac{c_1}{n} < |z| < 1 + \frac{c_1}{n} \right \}\,.$$中至少有$c_2n$零点
英文摘要：
In signal processing the Rudin-Shapiro polynomials have good autocorrelation properties and their values on the unit circle are small. Binary sequences with low autocorrelation coefficients are of interest in radar, sonar, and communication systems. In this paper we study the oscillation of the modulus of the Rudin-Shapiro polynomials on the unit circle. We also show that the Rudin-Shapiro polynomials $P_k$ and $Q_k$ of degree $n-1$ with $n := 2^k$ have $o(n)$ zeros on the unit circle. This should be compared with a result of B. Conrey, A. Granville, B. Poonen, and K. Soundararajan stating that for odd primes $p$ the Fekete polynomials $f_p$ of degree $p-1$ have asymptotically $κ_0 p$ zeros on the unit circle, where $0.500813>κ_0>0.500668$. Our approach is based heavily on the Saffari and Montgomery conjectures proved recently by B. Rodgers. We also prove that there are absolute constants $c_1 > 0$ and $c_2 > 0$ such that the $k$-th Rudin-Shapiro polynomials $P_k$ and $Q_k$ of degree $n-1 = 2^k-1$ have at least $c_2n$ zeros in the annulus $$\left \{z \in {\Bbb C}: 1 - \frac{c_1}{n} < |z| < 1 + \frac{c_1}{n} \right \}\,.$$
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
