Multifractal Analysis of generalized Thue-Morse trigonometric polynomials

TL;DR

This paper performs a detailed multifractal analysis of generalized Thue-Morse trigonometric polynomials, examining their pointwise behavior and limit properties, extending previous work on their uniform norms.

Contribution

It provides a comprehensive multifractal analysis of the pointwise behavior of generalized Thue-Morse polynomials, including the limit of their logarithmic magnitude.

Findings

Established the pointwise multifractal spectrum of the polynomials.

Derived the limit behavior of the logarithm of polynomial magnitudes.

Extended understanding of the multifractal structure of Thue-Morse sequences.

Abstract

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 the polynomials $\sigma_{N}^{(c)} (x) := \sum_{n=0}^{N-1} t_n^{(c)} e^{2\pi i n x}$, 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. In this paper, we study the pointwise behavior and give a complete multifractal analysis of the limit $\lim_{n\to\infty}n^{-1}\log |\sigma_{2^n}^{(c)}(x)|$.