Multifractal analysis of generalized Thue-Morse trigonometric polynomials

We consider the generalized Thue–Morse sequences $(t^{(c)}_n)_{n \geq 0}$ ($c \in [0, 1)$ being a parameter) defined by $t^{(c)}_n = e^{2 \pi ics_2(n)}$, where $s_2(n)$ is the sum of digits of the binary expansion of $n$. For the polynomials $\sigma^{(c)}_N (x) := \sum^{N-1}_{n=0} t^{(c)}_n e^{2 \pi inx}$, we have proved in $\href{https://doi.org/10.3934/dcds.2020363 }{[18]}$ that the uniform norm ${\lVert \sigma^{(c)}_N \rVert}_\infty$ 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 $\lim_{n \to \infty} n^{-1} \log {\lvert \sigma^{(c)}_{2^n} (x) \rvert}$.

Keywords

generalized Thue-Morse sequences, Hausdorff dimension, multifractal analysis

2010 Mathematics Subject Classification

Primary 37Axx. Secondary 28A80.

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Dedicated to the memory of Professor Ka-Sing Lau

Received 27 December 2022

Accepted 18 July 2023

Published 10 July 2024