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Structure algébrique sur les couples de Rudin-Shapiro. Problème extrémal de Salem sur les polynômes à coefficients $$\pm 1$$. (Algebraic structure on Rudin-Shapiro pairs. Salem’s extremal problem on polynomials with coefficients $$\pm 1)$$. (French) Zbl 0608.10052
For every integer n, let $$K_ n=\inf (n^{-1/2} \sup_{\theta}| \sum^{n-1}_{k=0}\eta_ k e^{ik\theta}|)$$, the infimum being taken over all the $$(\eta_ 0,...,\eta_{n-1})$$ such that $$\eta_ k=\pm 1$$, $$k=0,...,n-1$$. The author proves that $$\limsup_{n\to \infty}K_ n\leq \sqrt{2}$$ (actually he conjectures that equality holds, while Erdős has conjectured that $$\liminf_{n\to \infty}K_ n=1).$$
Let $${\mathcal R}$$ be the set of the pairs of polynomials (P,Q) whose coefficients are $$\pm 1$$, such that $$| P(e^{i\theta})|^ 2+| Q(e^{i\theta})|^ 2$$ be constant (when $$\theta$$ runs in $${\mathbb{R}})$$. The author puts on $${\mathcal R}$$ an internal law that makes it a non-commutative ”normed” semigroup with neutral element (1,1); the norm of a (P,Q) is (deg P)$$+1$$ (recall deg P$$=\deg Q)$$. Then every (P,Q) of degree $$\geq 1$$ has a factorization in ”prime factors” (of degree 1). That factorization together with the norm properties help the author to find the upper inequality.
Reviewer: A.Escassut

##### MSC:
 11B83 Special sequences and polynomials 11L40 Estimates on character sums 42A05 Trigonometric polynomials, inequalities, extremal problems 11K06 General theory of distribution modulo $$1$$ 12D10 Polynomials in real and complex fields: location of zeros (algebraic theorems)
##### Keywords:
Rudin-Shapiro sequences; Rudin-Shapiro pairs; inequality