On the distribution of Rudin–Shapiro polynomials and lacunary walks on SU(2)

Rodgers, Brad

doi:10.1016/j.aim.2017.09.022

Cited by 10 publications

(22 citation statements)

References 17 publications

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“…Then

P_{k_{j}}

has at least one zero on the unit circle and hence and imply that

false∥ P_{k_{j}} false(e^{i t} false) {false∥}_{K}^{2} = 2^{k_{j} + 1} .

Then Theorem implies that

m ((}{t \in K : false| P_{k_{j}} false(e^{i t} false) {false|}^{2} \leq α {∥ P_{k_{j}} ∥}_{K}^{2})) \geq \frac{\sqrt{α}}{e \sqrt{2}} \frac{ε n_{j}}{n_{j}} = \frac{ε \sqrt{α}}{e \sqrt{2}}

for every

α \in (0, 1)

and

j = 1, 2, \dots

Hence,

\underset{j \to \infty}{lim inf} m ((}{t \in K : false| P_{k_{j}} false(e^{i t} false) {false|}^{2} \leq α {∥ P_{k_{j}} false(e^{i t} false) ∥}_{K}^{2})) \geq \frac{ε \sqrt{α}}{e \sqrt{2}}

for every

α \in (0, 1)

. On the other hand, Conjecture proved in combined with imply that

\lim_{j \to \infty} m ()

…”

Section: Proofs Of the Theoremsmentioning

confidence: 94%

“…Proof of Theorem

I_{j} : = [\frac{false(2 j - 2 false) π}{n}, \frac{2 j π}{n}), j = 1, 2, \dots, n .

Let

γ : = \sin^{2} false(π / 8 false)

as before. By Saffari's Conjecture proved by Rodgers we have

m (false{t \in K : R_{k} (t) \leq γ n false}) > 2 π false(γ / 4 false)

for every sufficiently large n . Hence, with the notation

A : = {t \in K : R_{k} false(t false) \leq γ n},

there are at least

n γ / 4

distinct values of

j \in {1, 2, \dots, n}

such that

A \cap I_{j} \neq \emptyset

for every sufficiently large n .…”

Section: Proofs Of the Theoremsmentioning

confidence: 98%

“…Proof of Theorem Recalling , without loss of generality we may assume that

η \in (0, 2 γ)

. Let

I_{j} : = [\frac{false(2 j - 2 false) π}{n}, \frac{2 j π}{n}), j = 1, 2, \dots, n .

By Saffari's Conjecture proved by Rodgers we have

m (false{t \in K : R_{k} (t) \leq η n false}) > π false(1 - ε false) η

for every

η \in (0, 1)

ε > 0

, and sufficiently large

k \geq k_{η, ε}

. Hence, with the notation

A_{η} : = false{t \in K : R_{k} (t) \leq η n false},

there are at least

(1 - ε) η n / 2

distinct values of

j \in {1, 2, \dots, n}

such that

A_{η} \cap I_{j} \neq \emptyset

for every

η \in (0, 1)

and sufficiently large

k \geq k_{η, ε}

.…”

Section: Proofs Of the Theoremsmentioning

confidence: 99%

“…Let γ := sin 2 (π/8) as before. By Saffari's Conjecture 1.1 proved by Rodgers [53] we have m({t ∈ K : R k (t ) γ n}) > 2π (γ /4) for every sufficiently large n. Hence, with the notation…”

mentioning

confidence: 91%

“…for every α ∈ (0, 1). On the other hand, Conjecture 1.1 proved in [53] combined with (4.4) imply that…”

mentioning

confidence: 98%

See 4 more Smart Citations

On the Oscillation of the Modulus of the Rudin–shapiro Polynomials on the Unit Circle

Erdélyi

2019

Mathematika

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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 Pk and Qk of degree n−1 with n:=2k have o(n) zeros on the unit circle. This should be compared with a result of Conrey, Granville, Poonen and Soundararajan stating that for odd primes p the Fekete polynomials fp of degree p−1 have asymptotically κ0p zeros on the unit circle, where 0.500813>κ0>0.500668. Our approach is based heavily on the Saffari and Montgomery conjectures proved recently by Rodgers. We also prove that there are absolute constants c1>0 and c2>0 such that the kth Rudin–Shapiro polynomials Pk and Qk of degree n−1=2k−1 have at least c2n zeros in the annulus z∈double-struckC:1−c1n<|z|<1+c1n.

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“…Then

P_{k_{j}}

has at least one zero on the unit circle and hence and imply that

false∥ P_{k_{j}} false(e^{i t} false) {false∥}_{K}^{2} = 2^{k_{j} + 1} .

Then Theorem implies that

m ((}{t \in K : false| P_{k_{j}} false(e^{i t} false) {false|}^{2} \leq α {∥ P_{k_{j}} ∥}_{K}^{2})) \geq \frac{\sqrt{α}}{e \sqrt{2}} \frac{ε n_{j}}{n_{j}} = \frac{ε \sqrt{α}}{e \sqrt{2}}

for every

α \in (0, 1)

and

j = 1, 2, \dots

Hence,

\underset{j \to \infty}{lim inf} m ((}{t \in K : false| P_{k_{j}} false(e^{i t} false) {false|}^{2} \leq α {∥ P_{k_{j}} false(e^{i t} false) ∥}_{K}^{2})) \geq \frac{ε \sqrt{α}}{e \sqrt{2}}

for every

α \in (0, 1)

. On the other hand, Conjecture proved in combined with imply that

\lim_{j \to \infty} m ()

…”

Section: Proofs Of the Theoremsmentioning

confidence: 94%

“…Proof of Theorem

I_{j} : = [\frac{false(2 j - 2 false) π}{n}, \frac{2 j π}{n}), j = 1, 2, \dots, n .

Let

γ : = \sin^{2} false(π / 8 false)

as before. By Saffari's Conjecture proved by Rodgers we have

m (false{t \in K : R_{k} (t) \leq γ n false}) > 2 π false(γ / 4 false)

for every sufficiently large n . Hence, with the notation

A : = {t \in K : R_{k} false(t false) \leq γ n},

there are at least

n γ / 4

distinct values of

j \in {1, 2, \dots, n}

such that

A \cap I_{j} \neq \emptyset

for every sufficiently large n .…”

Section: Proofs Of the Theoremsmentioning

confidence: 98%

“…Proof of Theorem Recalling , without loss of generality we may assume that

η \in (0, 2 γ)

. Let

I_{j} : = [\frac{false(2 j - 2 false) π}{n}, \frac{2 j π}{n}), j = 1, 2, \dots, n .

By Saffari's Conjecture proved by Rodgers we have

m (false{t \in K : R_{k} (t) \leq η n false}) > π false(1 - ε false) η

for every

η \in (0, 1)

ε > 0

, and sufficiently large

k \geq k_{η, ε}

. Hence, with the notation

A_{η} : = false{t \in K : R_{k} (t) \leq η n false},

there are at least

(1 - ε) η n / 2

distinct values of

j \in {1, 2, \dots, n}

such that

A_{η} \cap I_{j} \neq \emptyset

for every

η \in (0, 1)

and sufficiently large

k \geq k_{η, ε}

.…”

Section: Proofs Of the Theoremsmentioning

confidence: 99%

mentioning

confidence: 91%

“…for every α ∈ (0, 1). On the other hand, Conjecture 1.1 proved in [53] combined with (4.4) imply that…”

mentioning

confidence: 98%

See 3 more Smart Citations

On the Oscillation of the Modulus of the Rudin–shapiro Polynomials on the Unit Circle

Erdélyi

2019

Mathematika

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The asymptotic value of the Mahler measure of the Rudin-Shapiro polynomials

Erdélyi

2020

JAMA

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Distribution of the zeros of polynomials near the unit circle

Das

2024

Journal of Approximation Theory

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On the distribution of Rudin–Shapiro polynomials and lacunary walks on SU(2)

Cited by 10 publications

References 17 publications

On the Oscillation of the Modulus of the Rudin–shapiro Polynomials on the Unit Circle

On the Oscillation of the Modulus of the Rudin–shapiro Polynomials on the Unit Circle

The asymptotic value of the Mahler measure of the Rudin-Shapiro polynomials

Distribution of the zeros of polynomials near the unit circle

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