\begin{document}$ L^\infty $\end{document} -estimation of generalized Thue-Morse trigonometric polynomials and ergodic maximization

Given an integer \begin{document}$ q\ge 2 $\end{document} and a real number \begin{document}$ c\in [0,1) $\end{document} , consider the generalized Thue-Morse sequence \begin{document}$ (t_n^{(q;c)})_{n\ge 0} $\end{document} defined by \begin{document}$ t_n^{(q;c)} = e^{2\pi i c s_q(n)} $\end{document} , where \begin{document}$ s_q(n) $\end{document} is the sum of digits of the \begin{document}$ q $\end{document} -expansion of \begin{document}$ n $\end{document} . We prove that the \begin{document}$ L^\infty $\end{document} -norm of the trigonometric polynomials \begin{document}$ \sigma_{N}^{(q;c)} (x) : = \sum_{n = 0}^{N-1} t_n^{(q;c)} e^{2\pi i n x} $\end{document} , behaves like \begin{document}$ N^{\gamma(q;c)} $\end{document} , where \begin{document}$ \gamma(q;c) $\end{document} is equal to the dynamical maximal value of \begin{document}$ \log_q \left|\frac{\sin q\pi (x+c)}{\sin \pi (x+c)}\right| $\end{document} relative to the dynamics \begin{document}$ x \mapsto qx \mod 1 $\end{document} and that the maximum value is attained by a \begin{document}$ q $\end{document} -Sturmian measure. Numerical values of \begin{document}$ \gamma(q;c) $\end{document} can be computed.