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

Erdélyi, Tamás

doi:10.1007/s11854-020-0142-3

Cited by 3 publications

(3 citation statements)

References 22 publications

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“…Lemma 3.1 tells us that the modulus of the Rudin-Shapiro polynomials P k is certainly not smaller than (2γn) 1/2 at least at one of any two consecutive n-th root of unity, where n := 2 k . This is a crucial observation proved in [11] and plays a key role in [12], [13], [14] and [15] as well. Our Lemma 3.2 below follows from Lemma 3.1 reasonably simply.…”

Section: Lemmasmentioning

confidence: 66%

“…In [13] the asymptotic values of M 0 (P k , [0, 2π]) and M 0 (P k , [0, 2π]), conjectured by Saffari, have been found. Namely in [13] we showed the following.…”

Section: Theorem 13mentioning

confidence: 88%

“…See also [7] on sums of monomials with large Mahler measure on subarcs of the unit circle ∂D. In [13] the asymptotic values of M 0 (P k , [0, 2π]) and M 0 (P k , [0, 2π]), conjectured by Saffari, have been found. Namely in [13] we showed the following.…”

Section: Theorem 13mentioning

confidence: 97%

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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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Section: Lemmasmentioning

confidence: 66%

“…In [13] the asymptotic values of M 0 (P k , [0, 2π]) and M 0 (P k , [0, 2π]), conjectured by Saffari, have been found. Namely in [13] we showed the following.…”

Section: Theorem 13mentioning

confidence: 88%

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

Erdélyi

2019

Mathematika

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Do Flat Skew-Reciprocal Littlewood Polynomials Exist?

Erdélyi¹

2022

Constr Approx

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Improved results on the oscillation of the modulus of the Rudin-Shapiro polynomials on the unit circle

Erdélyi

2023

Proc. Amer. Math. Soc.

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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 . In a recent paper we combined close to sharp upper bounds for the modulus of the autocorrelation coefficients of the Rudin-Shapiro polynomials with a deep theorem of Littlewood to prove that there is an absolute constant A > 0 such that the equation R k (t) = (1 + η)n has at least An 0.5394282 distinct zeros in [0, 2π) whenever η is real, |η| < 2 −11 , and n is sufficiently large. In this paper we show that the equation R k (t) = (1 + η)n has at least (1/2 − |η| − ε)n/2 distinct zeros in [0, 2π) for every η ∈ (−1/2, 1/2), ε > 0, and sufficiently large k ≥ k η,ε .

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

Cited by 3 publications

References 22 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

Do Flat Skew-Reciprocal Littlewood Polynomials Exist?

Improved results on the oscillation of the modulus of the Rudin-Shapiro polynomials on the unit circle

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