Multifractal Analysis of generalized Thue-Morse trigonometric polynomials

We consider the generalized Thue-Morse sequences $(t_n^{(c)}{)}_{n\ge 0}$ ($c\in [0,1)$ being a parameter) defined by $t_n^{(c)}=e^{2\pi ic{s}_2(n)}$, where $s_2(n)$ is the sum of digits of the binary expansion of $n$. For the polynomials ${\sigma }_N^{(c)}(x):={\sum }_{n=0}^{N-1}{t}_n^{(c)}{e}^{2\pi inx}$, we have proved in [18] that the uniform norm $\mathrm{\parallel }{\sigma }_N^{(c)}{\mathrm{\parallel }}_{\mathrm{\infty }}$ behaves like $N^{\gamma (c)}$ and the best exponent $\gamma (c)$ is computed. In this paper, we study the pointwise behavior and give a complete multifractal analysis of the limit ${\lim }_{n\to \mathrm{\infty }}{n}^{-1}\log \mathrm{\mid }{\sigma }_{2^n}^{(c)}(x)\mathrm{\mid }$.