On the oscillation of the modulus of the Rudin-Shapiro polynomials on the unit circle

TL;DR

This paper investigates the oscillation and zero distribution of Rudin-Shapiro polynomials on the unit circle, revealing they have fewer zeros on the circle than Fekete polynomials and many zeros near it.

Contribution

It provides new bounds on the number of zeros of Rudin-Shapiro polynomials on and near the unit circle, extending understanding of their oscillatory behavior.

Findings

Rudin-Shapiro polynomials have o(n) zeros on the unit circle.

At least c2n zeros lie within a narrow annulus around the unit circle.

The oscillation of the modulus on the circle is characterized.

Abstract

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 $\kappa_0 p$ zeros on the unit circle, where $0.500813>\kappa_0>0.500668$. 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 $$\left \{z \in {\Bbb C}: 1 - \frac{c_1}{n} < |z| < 1 + \frac{c_1}{n} \right \}\,.$$

Equations

$$\left\{z\in \mathbb{C}:1-\frac{c_1}{n}<|z|<1+\frac{c_1}{n}\right\}.$$

$$M_0(f):=\exp \left(\frac{1}{2\pi }{\int }_0^{2\pi }\log |f(e^{it})|dt\right).$$

$$M_0(f)=\underset{q\to 0+}{\lim }{M}_q(f),$$

$$M_q(f):={\left(\frac{1}{2\pi }{\int }_0^{2\pi }{\left|f(e^{it})\right|}^{q}dt\right)}^{1/q},q>0.$$

$$M_{\infty }(f):=\underset{t\in [0,2\pi ]}{\max }|f(e^{it})|=\underset{q\to \infty }{\lim }{M}_q(f).$$

$$M_0(f)=|c|\prod _{j=1}^n\max \{1,|z_j|\}$$

$$f(z)=c\prod _{j=1}^n(z-z_j),c,z_j\in \mathbb{C}.$$

$$M_q(S):=M_q(f),0\le q\le \infty ,$$

$${\text{Cal}L}_n:=\left\{f:f(z)=\sum _{j=0}^n{a}_j{z}^j,a_j\in \{-1,1\}\right\}$$

$${\text{Cal}K}_n:=\left\{f:f(z)=\sum _{j=0}^n{a}_j{z}^j,a_j\in \mathbb{C},|a_j|=1\right\}$$

$$M_0(f)\le M_2(f)=\sqrt{n+1}$$

$$c_1\sqrt{n_k+1}\le |f_{n_k}(z)|\le c_2\sqrt{n_k+1},z\in \partial D,$$

$$\begin{array}{rl}{P}_0(z)& :=1,Q_0(z):=1,\\ P_{k+1}(z)& :=P_k(z)+z^{2^k}{Q}_k(z),\\ Q_{k+1}(z)& :=P_k(z)-z^{2^k}{Q}_k(z),\end{array}$$

$$|P_{k+1}(z){|}^2+|Q_{k+1}(z){|}^2=2(|P_k(z){|}^2+|Q_k(z){|}^2),z\in \partial D.$$

$$|P_k(z){|}^2+|Q_k(z){|}^2=2^{k+1}=2n,z\in \partial D.$$

$$Q_k(-z)=P_k^{\ast }(z)=z^{n-1}(P_k(1/z),z\in \partial D,$$

$$|Q_k(-z)|=|P_k(z)|,z\in \partial D.i$$

$$\underset{n\to \infty }{\lim }\frac{1}{2^{n+1}}\sum _{f\in {\text{Cal}L}_n}\frac{(M_q(f){)}^q}{n^{q/2}}=\Gamma \left(1+\frac{q}{2}\right).$$

$$\underset{n\to \infty }{\lim }\frac{1}{2^{n+1}}\sum _{f\in {\text{Cal}L}_n}\frac{M_q(f)}{n^{1/2}}={\left(\Gamma \left(1+\frac{q}{2}\right)\right)}^{1/q}$$

$$\underset{n\to \infty }{\lim }\frac{1}{2^{n+1}}\sum _{f\in {\text{Cal}L}_n}\frac{M_0(f)}{n^{1/2}}=e^{-\gamma /2},$$

$$\gamma :=\underset{n\to \infty }{\lim }\left(\sum _{k=1}^n\frac{1}{k}-\log n\right)=0.577215\dots$$

$$M_q(P_k)=M_q(Q_k)\sim \frac{2^{k+1)/2}}{(q/2+1{)}^{1/q}}$$

$$\begin{array}{rl}& \underset{k\to \infty }{\lim }m\left(\left\{t\in K:{\left|\frac{P_k(e^{it})}{\sqrt{2^{k+1}}}\right|}^2\in [\alpha ,\beta ]\right\}\right)\\ =& \underset{k\to \infty }{\lim }m\left(\left\{t\in K:{\left|\frac{Q_k(e^{it})}{\sqrt{2^{k+1}}}\right|}^2\in [\alpha ,\beta ]\right\}\right)=2\pi (\beta -\alpha )\end{array}$$

$$\begin{array}{rl}& \underset{k\to \infty }{\lim }m\left(\left\{t\in K:\frac{P_k(e^{it})}{\sqrt{2^{k+1}}}\in E\right\}\right)\\ =& \underset{k\to \infty }{\lim }m\left(\left\{t\in K:\frac{Q_k(e^{it})}{\sqrt{2^{k+1}}}\in E\right\}\right)=2m(E)\end{array}$$

$$\left\{z\in \mathbb{C}:1-\frac{c_1}{n}<|z|<1+\frac{c_1}{n}\right\},$$

$$f_p(z):=\sum _{j=1}^{p-1}\left(\frac{j}{p}\right)z^j,$$

$$\left(\frac{j}{p}\right)=\{\begin{array}{l}1,\text{if }{x}^2\equiv j(\text{mod}p)\text{has a nonzero solution,}\\ 0,\text{if }p\text{divides}j,\\ -1,\text{otherwise}\end{array}$$

$$\Vert g{\Vert }_A:=\underset{x\in A}{\sup }|g(x)|$$

$$m(\{t\in K:|S(t)|\le \alpha \Vert S{\Vert }_K\})\ge \frac{\sqrt{\alpha }}{e}\frac{k}{n}$$

$$\left\{z\in \mathbb{C}:1-\frac{c_1}{n}<|z|<1+\frac{c_1}{n}\right\}.$$

Full text

On the oscillation of the modulus of the Rudin-Shapiro polynomials on the unit circle

Tamás Erdélyi

Department of Mathematics, Texas A&M University, College Station, Texas 77843, College Station, Texas 77843

terdelyimath.tamu.edu

(February 28, 2018 )

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 $\kappa_{0}p$ zeros on the unit circle, where $0.500813>\kappa_{0}>0.500668$. 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 $c_{2}n$ zeros in the annulus

[TABLE]

polynomials, restricted coefficients, number of zeros on the unit circle, Rudin-Shapiro polynomials

††support: 2010 Mathematics Subject Classifications. 11C08, 41A17, 26C10, 30C15

1. Introduction and Notation

Let $D:=\{z\in{\mathbb{C}}:|z|<1\}$ denote the open unit disk of the complex plane. Let $\partial D:=\{z\in{\mathbb{C}}:|z|=1\}$ denote the unit circle of the complex plane. The Mahler measure $M_{0}(f)$ is defined for bounded measurable functions $f$ on $\partial D$ by

[TABLE]

It is well known, see [HL-52], for instance, that

[TABLE]

where

[TABLE]

It is also well known that for a function $f$ continuous on $\partial D$ we have

[TABLE]

It is a simple consequence of the Jensen formula that

[TABLE]

for every polynomial of the form

[TABLE]

See [BE-95, p. 271] or [B-02, p. 3], for instance. It will be convenient for us to introduce the notation

[TABLE]

for functions $S$ defined on the period $K:={\mathbb{R}}\enskip(\text{mod}\,\,2\pi)$ by $S(t):=f(e^{it})$, where $f$ is a bounded measurable functions $f$ on $\partial D$.

Let ${\Cal{P}}_{n}^{c}$ be the set of all algebraic polynomials of degree at most $n$ with complex coefficients. Let ${\Cal{T}}_{n}$ be the set of all real (that is, real-valued on the real line) trigonometric polynomials of degree at most $n$. Finding polynomials with suitably restricted coefficients and maximal Mahler measure has interested many authors. The classes

[TABLE]

of Littlewood polynomials and the classes

[TABLE]

of unimodular polynomials are two of the most important classes considered. Observe that ${\Cal{L}}_{n}\subset{\Cal{K}}_{n}$ and

[TABLE]

for every $f\in{\Cal{K}}_{n}$. Beller and Newman [BN-73] constructed unimodular polynomials $f_{n}\in{\Cal{K}}_{n}$ whose Mahler measure $M_{0}(f_{n})$ is at least $\sqrt{n}-c/\log n$.

Section 4 of [B-02] is devoted to the study of Rudin-Shapiro polynomials. Littlewood asked if there were polynomials $f_{n_{k}}\in{\Cal{L}}_{n_{k}}$ satisfying

[TABLE]

with some absolute constants $c_{1}>0$ and $c_{2}>0$, see [B-02, p. 27] for a reference to this problem of Littlewood. To satisfy just the lower bound, by itself, seems very hard, and no such sequence $(f_{n_{k}})$ of Littlewood polynomials $f_{n_{k}}\in{\Cal{L}}_{n_{k}}$ is known. A sequence of Littlewood polynomials that satisfies just the upper bound is given by the Rudin-Shapiro polynomials. The Rudin-Shapiro polynomials appear in Harold Shapiro’s 1951 thesis [S-51] at MIT and are sometimes called just Shapiro polynomials. They also arise independently in Golay’s paper [G-51]. They are remarkably simple to construct and are a rich source of counterexamples to possible conjectures.

The Rudin-Shapiro polynomials are defined recursively as follows:

[TABLE]

for $k=0,1,2,\ldots\,.$ Note that both $P_{k}$ and $Q_{k}$ are polynomials of degree $n-1$ with $n:=2^{k}$ having each of their coefficients in $\{-1,1\}$. 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.

It is well known and easy to check by using the parallelogram law that

[TABLE]

Hence

[TABLE]

It is also well known (see Section 4 of [B-02], for instance), that

[TABLE]

and hence

[TABLE]

P. Borwein’s book [B-02] presents a few more basic results on the Rudin-Shapiro polynomials. Various properties of the Rudin-Shapiro polynomials are discussed in [B-73] by Brillhart and in [BL-76] by Brillhart, Lemont, and Morton. Obviously $M_{2}(P_{k})=2^{k/2}$ by the Parseval formula. In 1968 Littlewood [L-68] evaluated $M_{4}(P_{k})$ and found that $M_{4}(P_{k})\sim(4^{k+1}/3)^{1/4}$. The $M_{4}$ norm of Rudin-Shapiro like polynomials on $\partial D$ are studied in [BM-00].

P. Borwein and Lockhart [BL-01] investigated the asymptotic behavior of the mean value of normalized $M_{q}$ norms of Littlewood polynomials for arbitrary $q>0$. They proved that

[TABLE]

In [C-15c] we proved that

[TABLE]

for every $q>0$. In [CE-15c] we also proved the following result on the average Mahler measure of Littlewood polynomials. We have

[TABLE]

where

[TABLE]

is the Euler constant and $e^{-\gamma/2}=0.749306\ldots$. These are analogues of the results proved earlier by Choi and Mossinghoff [CM-11] for polynomials in ${\Cal{K}}_{n}$. Let $K:={\mathbb{R}}\enskip(\text{mod}\,\,2\pi)$. Let $m(A)$ denote the one-dimensional Lebesgue measure of $A\subset K$. In 1980 Saffari conjectured the following.

Conjecture 1.1

Let $P_{k}$ and $Q_{k}$ be the Rudin-Shapiro polynomials of degree $n-1$ with $n:=2^{k}$. We have

[TABLE]

for all real exponents $q>0$. Equivalently, we have

[TABLE]

whenever $0\leq\alpha<\beta\leq 1$.

This conjecture was proved for all even values of $q\leq 52$ by Doche [D-05] and Doche and Habsieger [DH-04]. Recently B. Rodgers [R-16] proved Saffari’s Conjecture 1.1 for all $q>0$. See also [EZ-17]. An extension of Saffari’s conjecture is Montgomery’s conjecture below.

Conjecture 1.2

Let $P_{k}$ and $Q_{k}$ be the Rudin-Shapiro polynomials of degree $n-1$ with $n:=2^{k}$. We have

[TABLE]

for any measurable set $E\subset D:=\{z\in{\mathbb{C}}:|z|<1\}\,.$

B. Rodgers [R-16] proved Montgomery’s Conjecture 1.2 as well.

Despite the simplicity of their definition not much is known about the Rudin-Shapiro polynomials. It has been shown in [E-16c] fairly recently that the Mahler measure ($M_{0}$ norm) and the $M_{\infty}$ norm of the Rudin-Shapiro polynomials $P_{k}$ and $Q_{k}$ of degree $n-1$ with $n:=2^{k}$ on the unit circle of the complex plane have the same size, that is, the Mahler measure of the Rudin-Shapiro polynomials of degree $n-1$ with $n:=2^{k}$ is bounded from below by $cn^{1/2}$, where $c>0$ is an absolute constant.

It is shown in this paper 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. 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 $c_{2}n$ zeros in the annulus

[TABLE]

while there is an absolute constant $c>0$ such that each of the functions $\text{\rm Re}(P_{k})$, $\text{\rm Re}(Q_{k})$, $\text{\rm Im}(P_{k})$, and $\text{\rm Im}(Q_{k})$ has at least $cn$ zeros on the unit circle. The oscillation of $R_{k}(t):=|P_{k}(e^{it})|^{2}$ and $R_{k}(t):=|Q_{k}(e^{it})|^{2}$ on the period $[0,2\pi)$ is also studied.

For a prime number $p$ the $p$-th Fekete polynomial is defined as

[TABLE]

where

[TABLE]

is the usual Legendre symbol. Since $f_{p}$ has constant coefficient [math], it is not a Littlewood polynomial, but $g_{p}$ defined by $g_{p}(z):=f_{p}(z)/z$ is a Littlewood polynomial of degree $p-2$. Fekete polynomials are examined in detail in [B-02], [CG-00], [E-11], [E-12], [E-17], [EL-07], and [M-80]. In [CE-15a] and [CE-15b] the authors examined the maximal size of the Mahler measure and the $L_{p}$ norms of sums of $n$ monomials on the unit circle as well as on subarcs of the unit circles. In the constructions appearing in [CE-15a] properties of the Fekete polynomials $f_{p}$ turned out to be quite useful. In [CG-00] B. Conrey, A. Granville, B. Poonen, and K. Soundararajan proved that for an odd prime $p$ the Fekete polynomial f_{p}(z)=\sum^{p-1}_{j=0}\big{(}{j\over p}\big{)}z^{j} (the coefficients are Legendre symbols) has $\sim\kappa_{0}p$ zeros on the unit circle, where $0.500813>\kappa_{0}>0.500668$. So Fekete polynomials are far from having only $o(p)$ zeros on the unit circle.

Mercer [M-06a] proved that if a Littlewood polynomial $f\in{\Cal{L}}_{n}$ of the form $f(z)=\sum_{j=0}^{n}{a_{j}z^{j}}$ is skew-reciprocal, that is, $a_{j}=(-1)^{j}a_{n-j}$ for each $j=0,1,\ldots,n$, then it has no zeros on the unit circle. However, by using different elementary methods it was observed in both [E-01] and [M-06a] that if a Littlewood polynomial $P$ of the form (1.1) is self-reciprocal, that is, $a_{j,n}=a_{n-j}$ for each $j=0,1,\ldots,n$, $n\geq 1$, then it has at least one zero on the unit circle.

There are many other papers on the zeros of constrained polynomials. Some of them are [BP-32], [BE-97], [BE-01], [BE-07], [BE-08a], [BE-08b], [BE-99], [BE-13], [B-97], [D-08], [E-08a], [E-08b], [E-16a], [E-16b], [L-61], [L-64], [L-66a], [L-66b], [L-68], [M-06b], [Sch-32], [Sch-33], [Sz-34], and [TV-07].

2. New Results

Let $P_{k}$ and $Q_{k}$ be the Rudin-Shapiro polynomials of degree $n-1$ with $n:=2^{k}$. Let either $R_{k}(t):=|P_{k}(e^{it})|^{2}$ or $R_{k}(t):=|Q_{k}(e^{it})|^{2}$. Let $\gamma:=\sin^{2}(\pi/8)$. We use the notation

[TABLE]

for a complex-valued function $g$ defined on a set $A\subset{\mathbb{R}}$.

Theorem 2.1

$P_{k}$ and $Q_{k}$ have $o(n)$ zeros on the unit circle.

The proof of Theorem 2.1 will follow by combining the recently proved Saffari’s conjecture stated as Conjecture 1.1 and the theorem below. Let $K:={\mathbb{R}}\enskip(\text{mod}\,\,2\pi)$.

Theorem 2.2

If $S\in{\Cal{T}}_{n}$ is of the form $S(t)=|f(e^{it})|^{2}$, where $f\in{\Cal{P}}_{n}^{c}$, and $f$ has at least $k$ zeros (counted with multiplicities) in $K$, then

[TABLE]

for every $\alpha\in(0,1)$, where $m(A)$ denotes the one-dimensional Lebesgue measure of a measurable set $A\subset K$.

Theorem 2.3

There is an absolute constant $c>0$ such that each of the functions $\text{\rm Re}(P_{k})$, $\text{\rm Re}(Q_{k})$, $\text{\rm Im}(P_{k})$, and $\text{\rm Im}(Q_{k})$ has at least $cn$ zeros on the unit circle for every $n=2^{k}-1\geq 1$.

Theorem 2.4

There is an absolute constant $c>0$ such that the equation $R_{k}(t)=\eta n$ has at most $c\eta^{1/2}n$ solutions (counted with multiplicities) in $K$ for every $\eta\in(0,1]$ and sufficiently large $k\geq k_{\eta}$, while the equation $R_{k}(t)=\eta n$ has at most $c(2-\eta)^{1/2}n$ solutions (counted with multiplicities) in $K$ for every $\eta\in[1,2)$ and sufficiently large $k\geq k_{\eta}$.

Theorem 2.5

The equation $R_{k}(t)=\eta n$ has at least $(1-\varepsilon)\eta n/2$ distinct solutions in $K$ for every $\eta\in(0,2\gamma)$, $\varepsilon>0$, and sufficiently large $k\geq k_{\eta,\varepsilon}$, while the equation $R_{k}(t)=\eta n$ has at least $(1-\varepsilon)(2-\eta)n/2$ distinct solutions in $K$ for every $\eta\in(2-\gamma,2)$, $\varepsilon>0$, and sufficiently large $k\geq k_{\eta,\varepsilon}$.

Theorem 2.6

There s an absolute constants $A>0$ such that the equation $R_{k}(t)=(1+\eta)n$ has at least $An^{0.36}$ distinct solutions in $K$ whenever $\eta$ is real and $|\eta|<2^{-11}$.

Theorem 2.7

There are absolute constants $c_{1}>0$ and $c_{2}>0$ such that $P_{k}$ and $Q_{k}$ have at least $c_{2}n$ zeros in the annulus

[TABLE]

We note that for every $c\in(0,1)$ there is an absolute constant $c_{3}>0$ depending only on $c$ such that every $U_{n}\in{\Cal{P}}_{n}^{c}$ of the form

[TABLE]

has at least $cn$ zeros in the annulus

[TABLE]

See Theorem 2.1 in [E-01].

On the other hand, there is an absolute constant $c_{3}>0$ such that for every $n\in{\mathbb{N}}$ there is a polynomial $U_{n}\in{\Cal{K}}_{n}$ having no zeros in the annulus (2.1). See Theorem 2.3 in [E-01]. So in Theorem 2.7 some special properties, in addition to being Littlewood polynomials, of the Rudin-Shapiro polynomials must be exploited.

A key to the proof of Theorem 2.7 is the result below.

Theorem 2.8

Let $t_{0}\in{\mathbb{R}}$. There is an absolute constant $c_{4}>0$ depending only on $c>0$ such that $P_{k}$ has at least one zero in the disk

[TABLE]

whenever

[TABLE]

Problem 2.9

Is there an absolute constant $c>0$ such that the equation $R_{k}(t)=\eta n$ has at least $c\eta n$ distinct solutions in $K$ for every $\eta\in(0,1)$ and sufficiently large $k\geq n_{\eta}$? In other words, can Theorem 2.5 be extended to all $\eta\in(0,1)$?

We note that it follows from $Q_{k}(z)=P_{k}^{*}(-z)$, $z\in{\mathbb{C}}$, that the products $P_{k}Q_{k}$ have at least $n-1$ zeros in the closed unit disk and at least $n-1$ zeros outside the open unit disk. So in the light of Theorem 2.1 the products $P_{k}Q_{k}$ have asymptotically $n$ zeros in the open unit disk. However, as far as we know, the following questions are open.

Problem 2.10

Is there an absolute constant $c>0$ such that $P_{k}$ has at least $cn$ zeros in the open unit disk?

Problem 2.11

Is there an absolute constant $c>0$ such that $Q_{k}$ has at least $cn$ zeros in the open unit disk?

Problem 2.12

Is it true that both $P_{k}$ and $Q_{k}$ have asymptotically half of their zeros in the open unit disk?

Problem 2.13

Is it true that if $n$ is odd then $P_{k}$ has a zero on the unit circle ${\partial D}$ only at $-1$ and $Q_{k}$ has a zero on the unit circle ${\partial D}$ only at $1$, while if $n$ is even then neither $P_{k}$ nor $Q_{k}$ has a zero on the unit circle?

As for $k\geq 1$ both $P_{k}$ and $Q_{k}$ have odd degree, both $P_{k}$ and $Q_{k}$ have at least one real zero. The fact that for $k\geq 1$ both $P_{k}$ and $Q_{k}$ have exactly one real zero was proved in [B-73].

3. Lemmas

Let, as before, $P_{k}$ and $Q_{k}$ be the Rudin-Shapiro polynomials of degree $n-1$ with $n:=2^{k}$. To prove Theorem 2.1 we need the lemma below that is proved in [BE-95, E.11 of Section 5.1 on pages 236–237].

Lemma 3.1

If $S\in{\Cal{T}}_{n}$, $t_{0}\in K$, and $r>0$, then $S$ has at most $enr|S(t_{0})|^{-1}\|S\|_{K}$ zeros in the interval $[t_{0}-r,t_{0}+r]$.

Our next lemma is in [E-18], and for the sake of completeness we present its short proof in

Our next lemma is stated as Lemma 3.5 in [E-16c], where its proof may also be found.

Lemma 3.2

If $\gamma:=\sin^{2}(\pi/8)$ and

[TABLE]

then

[TABLE]

for every $j=2u$, $u\in{\mathbb{Z}}$.

By Lemma 3.2, for every $n=2^{k}$ there are

[TABLE]

such that

[TABLE]

and with

[TABLE]

we have

[TABLE]

(Moreover, each $a_{j}$ is an $n$-th root of unity.)

Our next lemma is stated and proved as Lemma 3.4 in [E-18].

Lemma 3.3

There is an absolute constant $c_{3}>0$ such that

[TABLE]

for every sufficiently large $n=2^{k}\geq n_{\varepsilon}$, $k=1,2,\ldots$, and $\varepsilon>0$.

Our next lemma is based on the work of M. Taghavi [T-97] and gives an upper bound for the so-called autocorrelation coefficients of the Rudin-Shapiro polynomials.

Lemma 3.4

If

[TABLE]

($a_{0}=n$, $a_{j}=a_{-j}$, $j\geq 1$), then

[TABLE]

with an absolute constant $C>0$.

In fact, Taghavi [T-97] claimed

[TABLE]

However, as Allouche and Saffari observed, in his proof Taghavi used an incorrect statement saying that the spectral radius of the product of some matrices is independent of the order of the factors. So what he ended up with cannot be viewed as a correctly proved result. Building on what is correct in [T-97] Stephen Choi made some computations leading to the above correct form of Taghavi’s upper bound on the autocorrelation coefficients of the Rudin-Shapiro polynomials. The correction based on Choi’s computations will be the subject of a forthcoming note [AC-17] perhaps even in a more optimized form.

Using the notation of Lemma 3.4 Taghavi [T-96] claims also that

[TABLE]

for all $n:=2^{k}$ with an absolute constant $C>0$.

Our next lemma is due to Littlewood, see [Theorem 1 in L-66a].

Lemma 3.5

If $S\in{\Cal{T}}_{n}$ of the form

[TABLE]

satisfies

[TABLE]

where $c>0$ is a constant, $b_{0}=0$,

[TABLE]

for some constant $h>0$, and $v\in{\mathbb{R}}$ satisfies

[TABLE]

then

[TABLE]

where ${\Cal{N}}(S,v)$ denotes the number of real zeros of $S-v\mu$ in $(-\pi,\pi)$, and $A>0$ is an absolute constant.

Our next lemma is a key to prove Theorem 2.7. It is an extension of Theorem 1 in [E-02] establishing the right Bernstein inequality for trigonometric polynomials $S\in{\Cal{T}}_{n}$ not vanishing in the strip

[TABLE]

Lemma 3.6

Let $0<r\leq 2$. We have

[TABLE]

for every $S\in{\Cal{T}}_{n}$ having no zeros in the disk $D(a,r):=\{z\in{\mathbb{C}}:|z-a|<r\}$.

For the proof of Lemma 3.6 we need the lemma below.

Lemma 3.7

Let $2/n<r\leq 2$. We have

[TABLE]

for every $S\in{\Cal{T}}_{n}$ having no zeros in the disk $D(a,r)$ centered at $a$ of radius $r$, and for every $z$ in the square

[TABLE]

with $\rho:=\displaystyle{\frac{1}{5}\left(\frac{r}{2n}\right)^{1/2}}$.

For the proof of Lemma 3.7 we need the lemma below.

Lemma 3.8

Let $1/n\leq r\leq 1$. We have

[TABLE]

for every $S\in{\Cal{T}}_{n}$ having no zeros in the disk $D(x,r)$, and for every $y\in[-\rho,\rho]$ with $\rho:=\displaystyle{\frac{1}{5}\left(\frac{r}{n}\right)^{1/2}}$.

For the proof of Lemma 3.8 we need the lemma below stated as Lemma 4.3 in [E-98].

Lemma 3.9

Let $0<s\leq\lambda\leq 1$. We have

[TABLE]

for every $f\in{\Cal{P}}_{n}^{c}$ having no zeros in the disk $D(1-\lambda,\lambda)$ centered at $1-\lambda$ of radius $\lambda$.

Demonstration Proof of Lemma 3.8

It is sufficient to prove the lemma for $x=0$, since for $x\neq 0$ we can study the polynomial $\widetilde{S}\in{\Cal{T}}_{n}$ defined by $\widetilde{S}(t):=S(t-x)$ having no zeros in the disk $D(0,r)$. Associated with $S\in{\Cal{T}}_{n}$ we define $U\in{\Cal{T}}_{2n}$ by $U(t):=S(t)S(-t)$. Observe that $U$ is an even trigonometric polynomial of degree at most $2n$, hence we can define $f\in{\Cal{P}}_{2n}^{c}$ (in fact, with real coefficients) by

[TABLE]

Assume that $S$, and hence $U$, has no zeros in the disk $D(0,r)$. We show that $f$ has no zeros in the disk $D(1,2\lambda)$ centered at $1$ of radius $2\lambda:=r^{2}/4$. Indeed, as $S$, and hence $U$, has no zeros in the disk $D(0,r)$, $f$ has no zeros in the region $H:=\{u=\cos t:t\in D(0,r)\}$ bounded by the curve $\Gamma:=\partial H:=\{u=\cos t:|t|=r\}$. As $\cos 0=1$, $\Gamma$ goes around $1$ at least once by the Argument Principle. Observe that if $z=\cos t\in\Gamma$, then $|t|=r\leq 1$ implies that

[TABLE]

and hence $H$ contains the disk $D(1,2\lambda)=D(1,r^{2}/4)$. In conclusion, $f$ has no zeros in the disk $D(1,2\lambda)$ as we claimed.

Using Lemma 3.9 with $\displaystyle{\lambda:=\frac{r^{2}}{8}}$ and $s:=u-1$, we have

[TABLE]

Now let

[TABLE]

Then $1/n\leq r\leq 1$ and $\displaystyle{\lambda:=\frac{r^{2}}{8}}$ imply that

[TABLE]

Using (3.1) we have

[TABLE]

for every $y\in[-\rho,\rho]$ with $\rho:=\displaystyle{\frac{1}{5}\left(\frac{r}{n}\right)^{1/2}}$, $1/n\leq r\leq 1$. ∎

Demonstration Proof of Lemma 3.7

Observe that if $S\in{\Cal{T}}_{n}$ has no zeros in the disk $D(a,r)$ centered at $a$ of radius $r$, then it has no zeros in the disks $D(b,r/2)$ centered at $b\in[a-r/2,a+r/2]$ of radius $r/2$. Observe that $2/n\leq r\leq 2$ implies $1/n\leq r/2\leq 1$. Using Lemma 3.8 we obtain that

[TABLE]

for every $S\in{\Cal{T}}_{n}$ having no zeros in the disk $D(a,r)$ centered at $a$ of radius $r$, and for every

[TABLE]

with $\rho:=\displaystyle{\frac{1}{5}\left(\frac{r}{2n}\right)^{1/2}}$. As $2/n\leq r\leq 2$ implies $0<\rho<r/2$, the lemma follows. ∎

Demonstration Proof of Lemma 3.6

If $2/n\leq r\leq 2$ then using Cauchy’s integral formula and Lemma 3.7, we obtain

[TABLE]

for every $S\in{\Cal{T}}_{n}$ having no zeros in the disk $D(a,r):=\{z\in{\mathbb{C}}:|z-a|<r\}$. If $r<2/n$ then the classical Bernstein inequality valid for all $S\in{\Cal{T}}_{n}$ gives the lemma. ∎

4. Proofs of the Theorems

Demonstration Proof of Theorem 2.2

Let $S\in{\Cal{T}}_{n}$ be of the form $S(t)=|f(e^{it})|^{2}$, where $f\in{\Cal{P}}_{n}^{c}$. We define $U\in{\Cal{T}}_{n}$ and $V\in{\Cal{T}}_{n}$ by

[TABLE]

Then

[TABLE]

Suppose $S\in{\Cal{T}}_{n}$ defined by $S(t)=|f(e^{it})|^{2}$ has at least $u$ zeros in $K$, and let $\alpha\in(0,1)$. Then

[TABLE]

can be written as the union of pairwise disjoint intervals $I_{j}$, $j=1,2,\ldots,m$. Each of the intervals $I_{j}$ contains a point $y_{j}\in I_{j}$ such that

[TABLE]

Hence, (4.1) implies that for each $j=1,2,\ldots,m$, we have either

[TABLE]

or

[TABLE]

Also, each zero of $S$ lying in $K$ is contained in one of the intervals $I_{j}$. Let $\mu_{j}$ denote the number of zeros of $S$ lying in $I_{j}$. Since $S\in{\Cal{T}}_{n}$ has at least $u$ zeros in $K$, so do $U\in{\Cal{T}}_{n}$ and $V\in{\Cal{T}}_{n}$, and we have $\sum_{j=1}^{m}{\mu_{j}}\geq u$. Note that Lemma 3.1 applied to $U\in{\Cal{T}}_{n}$ yields that

[TABLE]

for each $j=1,2,\ldots,m$ for which (4.2) holds. Also, Lemma 3.1 applied to $V\in{\Cal{T}}_{n}$ yields that

[TABLE]

for each $j=1,2,\ldots,m$ for which (4.3) holds. Hence

[TABLE]

Therefore

[TABLE]

and the lemma follows. ∎

Demonstration Proof of Theorem 2.1

We show that the $P_{k}$ has $o(n)$ zeros on the unit circle, where $n=2^{k}-1$. The proof of the fact that $Q_{k}$ has $o(n)$ zeros on the unit circle is analogous. Suppose to the contrary that there are $\varepsilon>0$ and an increasing sequence $(k_{j})_{j=1}^{\infty}$ of positive integers such that the Rudin-Shapiro polynomials $P_{k_{j}}$ have at least $\varepsilon n_{j}$ zeros on the unit circle, where $n_{j}:=2^{k_{j}}$ for each $j=1,2,\ldots$. Then $P_{k_{j}}$ has at least one zero on the unit circle and hence (1.1) and (1.2) imply that

[TABLE]

Then Theorem 2.2 implies that

[TABLE]

for every $\alpha\in(0,1)$ and $j=1,2,\ldots$. Hence,

[TABLE]

for every $\alpha\in(0,1)$. On the other hand, Conjecture 1.1 proved in [R-16] combined with (4.4) imply that

[TABLE]

for every $\alpha\in(0,1)$. Combining (4.5) and (4.6) we obtain

[TABLE]

that is, $\varepsilon/e\leq 2\pi\sqrt{\alpha}$ for every $\alpha\in(0,1)$, a contradiction. ∎

Demonstration Proof of Theorem 2.3

We prove that there is an absolute constant $c>0$ such that $\text{\rm Re}(P_{k})$ has at least $cn$ zeros on the unit circle; the fact that each of the functions $\text{\rm Re}(Q_{k})$, $\text{\rm Im}(P_{k})$, and $\text{\rm Im}(Q_{k})$ has at least $cn$ zeros on the unit circle can be proved similarly. Let, as before, $K:={\mathbb{R}}\enskip(\text{mod}\,\,2\pi)$. Let

[TABLE]

Let $S\in{\Cal{A}}_{n-1}$ with $n:=2^{k}$ be defined by

[TABLE]

We have

[TABLE]

Let ${\Cal{N}}(S,v)$ be the number of real roots of $S-v\mu$ in $[-\pi,\pi)$. Observe that (1.1) implies that $|S(t)|\leq(2n)^{1/2}+1\leq 2(n-1)^{1/2}$ for all $t\in K$ and $n=2^{k}-1\geq 3$, and hence

[TABLE]

for all $n=2^{k}-1\geq 3$. Thus, applying Lemma 3.5 with $h=2^{9}c^{-6}$ we can deduce that there is an absolute constant $A>0$ such that

[TABLE]

has at least $A2^{-16}2^{-33/2}n=A2^{-65/2}n$ zeros in $[-\pi,\pi)$ whenever

[TABLE]

This finishes the proof when $n=2^{k}-1$ is sufficiently large. On the other hand, if $k\geq 1$ then $P_{k}$ always has at least one zero in the closed unit disk, hence $\text{\rm Re}(P_{k}(e^{it}))$ has at least two zeros in $[-\pi,\pi)$. ∎

Demonstration Proof of Theorem 2.4

The proof is a combination of Lemmas 3.1, 3.2, and 3.3. Recalling (1.2) we can observe that without loss of generality we may assume that $\eta\in(0,1]$, that is, it is sufficient to prove only the first statement of the theorem. As the trigonometric polynomial $R_{k}(t)-\eta n$ of degree $n-1$ has at most $2(n-1)$ zeros in $K$, without loss of generality we may assume also that $\eta<\gamma/2$, where $\gamma:=\sin^{2}(\pi/8)$ as before. In the light of Lemma 3.3 it is sufficient to prove that there is an absolute constant $c>0$ such that the equation $R_{k}(t)=\eta n$ has at most $c$ solutions in the interval $[\tau_{j-1},\tau_{j}]$ for every $j\in\{2,3,\ldots,m+1\}$ for which

[TABLE]

However, this follows from Lemmas 3.1 combined with Lemma 3.2. ∎

Demonstration Proof of Theorem 2.5

Recalling (1.1), without loss of generality we may assume that $\eta\in(0,2\gamma)$. Let

[TABLE]

By Saffari’s Conjecture 1.1 proved by Rodgers [R-16] we have

[TABLE]

for every $\eta\in(0,1)$, $\varepsilon>0$, and sufficiently large $k\geq k_{\eta,\varepsilon}$. Hence, with the notation

[TABLE]

there are at least $(1-\varepsilon)\eta n/2$ distinct values of $j\in\{1,2,\ldots,n\}$ such that $A_{\eta}\cap I_{j}\neq\emptyset$ for every $\eta\in(0,1)$ and sufficiently large $k\geq k_{\eta,\varepsilon}$. On the other hand, by Lemma 3.2, for each $j\in\{1,2,\ldots,n\}$ there is a $t_{j}\in I_{j}$ such that $R_{k}(t_{j})\geq 2\gamma n$. Hence by the Intermediate Value Theorem there are at least $(1-\varepsilon)\eta n/2$ distinct values of $j\in\{1,2,\ldots,n\}$ for which there is a $\tau_{j}\in I_{j}$ such that $R_{k}(\tau_{j})=\eta n$ for every $\eta\in(0,2\gamma)$, $\varepsilon>0$, and sufficiently large $k\geq k_{\eta,\varepsilon}$. ∎

Demonstration Proof of Theorem 2.6

Let $S_{n}\in{\Cal{T}}_{n-1}$ be defined by

[TABLE]

We show that $S:=S_{n}$ satisfies the assumptions of Lemma 3.5 with $c=1/4$ and $h:=n^{0.64}$ if $n=2^{k}$ is sufficiently large. Clearly, $S_{n}$ is of the form (3.1) with $b_{0}=0$, $b_{m}=a_{m}/2$, and $\gamma_{m}=0$ for $m=1,2,\ldots,n-1$. As it is already mentioned in Section 1, Littlewood [L-68] evaluated $M_{4}(P_{k})$ and found that $M_{4}(P_{k})\sim(4^{k+1}/3)^{1/4}=(4n^{2}/3)^{1/4}$. Hence $\mu:=M_{2}(S_{n})\sim(1/3)^{1/2}n$. Also, it follows from (1.1) that $M_{\infty}(S_{n})\leq n$, hence

[TABLE]

implies that $M_{1}(S_{n})\geq(1/4)n$ if $n=2^{k}$ is sufficiently large. Now Lemma 3.4, $b_{0}=0$, $b_{m}=2a_{m}\in\{-2,2\}$, $m=1,2,\ldots,n-1$, and $h:=n^{0.64}$ imply that

[TABLE]

if $n=2^{k}$ is sufficiently large. So $S_{n}$ satisfies the assumptions of Lemma 3.5 with $c=1/4$ and $h:=n^{0.64}$ if $n=2^{k}$ is sufficiently large, indeed. Thus Lemma 3.5 implies that

[TABLE]

whenever $v$ is real with $|v|\leq 2^{-5}c^{3}=2^{-11}$ and $n=2^{k}$ is sufficiently large. ∎

Demonstration Proof of Theorem 2.8

Suppose $P_{k}$ does not have a zero in the disk

[TABLE]

Observe that

[TABLE]

implies that $R_{k}\in{\Cal{T}}_{n}$ defined by $R_{k}(t)=P_{k}(e^{it})P_{k}(e^{-it})$ does not have a zero in

[TABLE]

It follows from Lemma 3.6 and $\|R_{k}\|_{K}\leq 2n$ that

[TABLE]

whenever

[TABLE]

Hence, if we chose $c_{4}>0$ as above, $P_{k}$ must have a zero in the disk

[TABLE]

whenever $R_{k}^{\prime}(t_{0})\geq cn^{2}$. ∎

Demonstration Proof of Theorem 2.7

[TABLE]

Let $\gamma:=\sin^{2}(\pi/8)$ as before. By Saffari’s Conjecture 1.1 proved by Rodgers [R-16] we have

[TABLE]

for every sufficiently large $n$. Hence, with the notation

[TABLE]

there are at least $n\gamma/4$ distinct values of $j\in\{1,2,\ldots,n\}$ such that $A\cap I_{j}\neq\emptyset$ for every sufficiently large $n$. On the other hand, by Lemma 3.2, for each $j\in\{1,2,\ldots,n\}$ there is a $t_{j}\in I_{j}$ such that $R_{k}(t_{j})\geq 2\gamma n$. Hence by the Mean Value Theorem there are at least $n\gamma/4$ distinct values of $j\in\{1,2,\ldots,n\}$ for which there is a $\tau_{j}\in I_{j}$ such that

[TABLE]

for every sufficiently large $n$. Hence, by Theorem 2.8, there are at least $n\gamma/4$ distinct values of $j\in\{1,2,\ldots,n\}$ such that the open disk $D_{j}$ centered at $e^{i\tau_{j}}$ of radius $c_{4}n^{-1}$ has at least one zero of $P_{k}$, where the absolute constant $c_{4}>0$ is chosen to $c:=\gamma/(2\pi)$ as in the proof of Theorem 2.8, that is,

[TABLE]

∎

5. Acknowledgement

The author thanks Stephen Choi for checking the details of the proof in this paper and for his computations leading to a correctly justified upper bound, stated as Lemma 3.4, for the autocorrelation coefficients of the Rudin-Shapiro polynomials.