On Polynomial Extractions of the Rudin–shapiro Sequence
نویسنده
چکیده
Let P (x) ∈ Z[x] be an integer-valued polynomial taking only positive values and let d be any fixed positive integer. The aim of this short note is to show, by elementary means, that for any sufficiently large integer N ≥ N0(P, d) there exists n such that P (n) contains exactly N occurrences of the block (q − 1, q − 1, . . . , q − 1) in its digital expansion in base q. The method of proof is constructive. It allows to give a lower estimate on the number of “0” resp. “1” symbols in polynomial extractions of the Rudin–Shapiro sequence.
منابع مشابه
The Mahler Measure of the Rudin-shapiro Polynomials
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 h...
متن کاملCrosscorrelation of Rudin-Shapiro-Like Polynomials
We consider the class of Rudin-Shapiro-like polynomials, whose L norms on the complex unit circle were studied by Borwein and Mossinghoff. The polynomial f(z) = f0 + f1z + · · ·+ fdz d is identified with the sequence (f0, f1, . . . , fd) of its coefficients. From the L 4 norm of a polynomial, one can easily calculate the autocorrelation merit factor of its associated sequence, and conversely. I...
متن کاملMoments of the Rudin-Shapiro Polynomials
We develop a new approach of the Rudin-Shapiro polynomials. This enables us to compute their moments of even order q for q 32, and to check a conjecture on the asymptotic behavior of these moments for q even and q 52.
متن کاملEven moments of generalized Rudin-Shapiro polynomials
We know from Littlewood (1968) that the moments of order 4 of the classical Rudin–Shapiro polynomials Pn(z) satisfy a linear recurrence of degree 2. In a previous article, we developed a new approach, which enables us to compute exactly all the moments Mq(Pn) of even order q for q 32. We were also able to check a conjecture on the asymptotic behavior of Mq(Pn), namely Mq(Pn) ∼ Cq2, where Cq = 2...
متن کاملGeneralised Rudin-Shapiro Constructions
A Golay Complementary Sequence (CS) has Peak-to-Average-Power-Ratio (PAPR) ≤ 2.0 for its one-dimensional continuous Discrete Fourier Transform (DFT) spectrum. Davis and Jedwab showed that all known length 2m CS, (GDJ CS), originate from certain quadratic cosets of Reed-Muller (1,m). These can be generated using the Rudin-Shapiro construction. This paper shows that GDJ CS have PAPR ≤ 2.0 under a...
متن کامل