Average Mahler’s measure and L p norms of unimodular polynomials

2017

Constr Approx

We show that there is an absolute constant c > 1/2 such that the Mahler measure of the Fekete polynomials f p of the form(where the coefficients are the usual Legendre symbols) is at least c √ p for all sufficiently large primes p. This improves the lower bound 1 2 − ε √ p known before for the Mahler measure of the Fekete polynomials f p for all sufficiently large primes p ≥ c ε . Our approach is based on the study of the zeros of the Fekete polynomials on the unit circle.

Section: Introduction and Notationsupporting

confidence: 80%

Improved Lower Bound for the Mahler Measure of the Fekete Polynomials

2017

Constr Approx

“…is the Euler constant. These are analogues of the results proved earlier by Choi and Mossinghoff [21] for polynomials in K n . Let m(A) denote the one-dimensional Lebesgue measure of A ⊂ K := R (mod 2π ).…”

supporting

confidence: 82%

“…We have

\lim_{n \to \infty} \frac{1}{2^{n + 1}} \sum_{f \in L_{n}} \frac{M_{0} (f)}{n^{1 / 2}} = e^{- γ / 2} = 0.749306 \dots,

where

γ : = \lim_{n \to \infty} (\sum_{k = 1}^{n} \frac{1}{k} - \log n) = 0.577215 \dots

is the Euler constant. These are analogues of the results proved earlier by Choi and Mossinghoff for polynomials in

K_{n}

. Let

m (A)

denote the one‐dimensional Lebesgue measure of

A \subset K : = R (\mod 2 π)

.…”

Section: Introduction and Notationsupporting

confidence: 67%

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

2019

Mathematika

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.

“…An analogue of this result does not seem to be known for q = 0 (the Mahler measure). However, the recent paper [9] paper establishes beautiful results on the average Mahler measure and L q (∂D) norms of unimodular polynomials (polynomials with complex coefficients of modulus 1) of degree n. 3…”

Section: Introductionmentioning

confidence: 99%

The Mahler Measure of the Rudin–Shapiro Polynomials

2015

Constr Approx

Abstract. Littlewood polynomials are polynomials with each of their coefficients in {−1, 1}. A sequence of Littlewood polynomials that satisfies a remarkable flatness property on the unit circle of the complex plane is given by the Rudin-Shapiro polynomials. It is shown in this paper that the Mahler measure and the maximum modulus of the Rudin-Shapiro polynomials on the unit circle of the complex plane have the same size. It is also shown that the Mahler measure and the maximum norm of the Rudin-Shapiro polynomials have the same size even on not too small subarcs of the unit circle of the complex plane. Not even nontrivial lower bounds for the Mahler measure of the Rudin Shapiro polynomials have been known before.