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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

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)