On the real roots of generalized Thue-Morse polynomials (Q2759135)

\textit{P. Erdős} and \textit{A. Offord} [Proc. Lond. Math. Soc. (3) 6, 139-160 (1956; Zbl 0070.01702)] proved that the mean value of real roots of a random polynomial of degree \(n\) with coefficients \(\pm 1\) is \((2/\pi)\log n\). The author and \textit{M. Mendès-France} [Exp. Math. 9, 339-350 (2000; Zbl 0976.52005)] proved that if we take the Thue-Morse sequence \((\varepsilon_i=(-1)^{\nu(i)})_{i\in{\mathbb N}}\), where \(\nu(i)\) is the sum of the binary digits of \(i\), then in the corresponding sequence of polynomials \(\sum_{i=0}^n \varepsilon_iX^i\) the average number of real zeros is \(11/4\).NEWLINENEWLINENEWLINEIn the paper under review the author defines by means of iterations of morphisms on words on the alphabet \(\{+,-\}\) induced by a word \(w\) the generalized Thue-Morse sequences \((\varepsilon_{w,i})_{i\in{\mathbb N}}\) for which the corresponding polynomials \(\sum_{i=0}^n \varepsilon_{w,i}X^i\) have for large \(n\) at least \(C\log n\) real roots, where \(C\) is an explicitly given constant.NEWLINENEWLINENEWLINEIn the final section the spectral measure of sequences \((\varepsilon_{w,i})_{i\in{\mathbb N}}\) is discussed. This is continuous, but singular (that is not absolutely continuous), consequently \((\varepsilon_{w,i})_{i\in{\mathbb N}}\) is pseudo-random in the sense of \textit{J. Bass} [Bull. Soc. Math. Fr. 87, 1-64 (1959; Zbl 0092.33404)] and \textit{J. Bertrandias} [ibid., Suppl., Mém. 5 (1966; Zbl 0148.11701)].