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On an extremal property of the Rudin-Shapiro sequence. (English) Zbl 0561.10025

W. Rudin and H. S. Shapiro have constructed a sequence \((\epsilon_ n)\) of \(\pm 1's\) such that for all \(N\geq 1\), and all \(\theta\) in [0,2\(\pi\) [: \[ | \sum^{N-1}_{n=0}\epsilon_ n \exp (2i\pi n\theta)| \leq (2+\sqrt{2}) \sqrt{N}. \] A unimodular 2-multiplicative sequence f is defined by \(| f(n)| =1\), for every integer n, \(f(a2^ n+b)=f(a2^ n) f(b)\), \(\forall n\geq 0\), \(\forall a\geq 1\), \(\forall b\) \(0\leq b<2^ n\). We prove that for all such sequences \[ | \sum^{N- 1}_{n=0}\epsilon_ n f(n)| \leq (2+\sqrt{2}) \sqrt{N}. \] This holds true in particular when f is the celebrated Thue-Morse sequence.

MSC:

11B83 Special sequences and polynomials
11L99 Exponential sums and character sums
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