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Multifractal Analysis of generalized Thue-Morse trigonometric polynomials

Aihua FanJ\"org SchmelingWeixiao Shen
Dec 2022
摘要
We consider the generalized Thue-Morse sequences $(t_n^{(c)})_{n\ge 0}$ ($c\in [0,1)$ being a parameter) defined by $t_n^{(c)} = e^{2\pi i c s_2(n)}$,where $s_2(n)$ is the sum of digits of the binary expansion of $n$. For thepolynomials $\sigma_{N}^{(c)} (x) := \sum_{n=0}^{N-1} t_n^{(c)} e^{2\pi i nx}$, we have proved in [18] that the uniform norm $\|\sigma_N^{(c)}\|_\infty$behaves like $N^{\gamma(c)}$ and the best exponent $\gamma(c)$ is computed. Inthis paper, we study the pointwise behavior and give a complete multifractalanalysis of the limit $\lim_{n\to\infty}n^{-1}\log |\sigma_{2^n}^{(c)}(x)|$.
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