Mixing of Prouhet–Thue–Morse and Rudin–Shapiro sequences

Abstract. Let \(P_\gamma(N,z)=\sum_{0\le n< N}z^n.\gamma(n)\) be a \(\mathbb C^2\)-valued polynomial where \(\gamma(\cdot)\) belongs to a family of sequences in \(\mathbb C^2\) derived from linear automata that recognize classical
Prouhet–Thue–Morse sequence and Rudin–Shapiro sequence in \(\{-1,+1\}\). Upper bound on the complex unit disk of the quadratic norms of \(P_\gamma(N,z)\) and analogous polynomials, obtained by summing on the intervals \([b,b+N)\), are estimated. Examples of applications and some generalizations are considered.