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

Abstract

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 $L^{\infty }$-norm of the trigonometric polynomials ${\sigma }_N^{(q;c)}(x):={\sum }_{n=0}^{N-1}{t}_n^{(q;c)}{e}^{2\pi inx}$, behaves like $N^{\gamma (q;c)}$, where $\gamma (q;c)$ is equal to the dynamical maximal value of ${\log }_q\left|\frac{\sin q\pi (x+c)}{\sin \pi (x+c)}\right|$ relative to the dynamics $x\mapsto qx\mathrm{mod}1$ and that the maximum value is attained by a $q$-Sturmian measure. Numerical values of $\gamma (q;c)$ can be computed.

38 pages, 7 figures, 2 tables