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

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