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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)
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