Note on an extremal property of the Rudin-Shapiro sequence (Q1913334)

[For part III, see the paper reviewed below.] Define \(u(n)\) to be the number of occurrences of the block 11 in the binary expansion of \(n\). The sequence \(((- 1)^{u(n)} )_n\) is the Rudin-Shapiro sequence. It satisfies \[ \sup_{\theta\in [0, 1]} \biggl|\sum_{n< N} (- 1)^{u(n)} e^{2i \pi n\theta} \biggr|\leq (2+ \sqrt {2} )\sqrt {N}. \] \textit{M. Mendès France} and the reviewer showed [Mathematika 32, 33-38 (1985; Zbl 0561.10025)] that, for any 2-multiplicative sequence \(f\) of modulus 1 (note that the assumption on the modulus of \(f\) is only implicit in the paper under review), one has \[ \biggl|\sum_{n< N} e^{2i\pi x u(n)} f(n) \biggr|\leq C(x) N^{\alpha (x)}, \tag \(*\) \] with \(\alpha (x)= \log (2+ 2|\cos \pi x|)/ \log 4\) and \(C(x)= \sqrt {2}/( \sqrt {2+ 2|\cos \pi x|} -1)\). The author shows that the term \(C(x) N^{\alpha (x)}\) in the inequality \((*)\) can be replaced by \(C(x) N^{\alpha (x)}- (C(x)- \sqrt {2})\). He also obtains the best possible constant \(B_n (\lambda, p)\) for which \(\sum^n_{k=1} x_k\leq B_n (\lambda, p) (\sum^n_{k=1} x^p_k )^{1/p}\), where \(p> 1\) and the real numbers \(x_k\) satisfy \(x_{k+1}\geq \lambda x_k> 0\) for a \(\lambda> 1\).