On the absolute value of the autocorrelations of the Thue-Morse sequence

TL;DR

This work addresses the growth rate of the absolute autocorrelations of the Thue–Morse sequence by embedding $\eta(m)$ in the framework of $k$-regular sequences and their linear representations. The authors develop a twisting approach using $2$-regular sign sequences and Kronecker-product structure to produce a lower bound, verifying a dominant-eigenvalue condition up to $n=25$ and establishing $\sum_{m\le x}|\eta(m)| \ge c_1 x^{0.6274882485}$ for large $x$. They also derive a sharp upper bound by extending recursions to arithmetic progressions and analyzing the associated matrices ${\bf R}_n$, obtaining $\sum_{m\le x}|\eta(m)|=O(x^{\alpha})$ with $\alpha<0.6464616660609581$ (verified up to $n=30$). Consequently, the optimal exponent lies in the interval $[0.6274882485,0.6464616661]$, with connections to the correlation dimension $D_2\approx 0.64298136$ and related Rényi dimensions. The results illuminate the fractal-like growth of these correlations and provide structural tools for studying products of regular sequences and their implications for spectral measures.

Abstract

Recently, Baake and Coons proved several results on the average size of the autocorrelations of the Thue--Morse sequence. They also considered the absolute value of the autocorrelations, and showed that the average value of the autocorrelations is zero. In particular, they showed that $\sum_{n\leqslant x}|η(n)|=o(x^α)$ for any $α>\log(3)/\log(4)$. In this paper, we sharpen this result, providing upper and lower bounds for $α$. On the way to our lower bounds, we obtain the structure of the linear representation of the point-wise product of two $k$-regular sequences, which may be of independent interest.

Figures (2)

• Figure 1: The values of $\sum_{m\leqslant x}f(m)\eta(m)$ for $f$ the paperfolding sequence (left), the Rudin--Shapiro sequence (middle), and the Thue--Morse sequence (right), for values of $x=0,\ldots,2^{13}-1$.
• Figure 2: The values of $x^{-1/2}\sum_{m\leqslant x}t(m)\eta(m)$ for $x=0,\ldots,2^{13}$.

Theorems & Definitions (14)

• Theorem 1.1
• Proposition 2.1: Dumas, 2013
• Proposition 3.1
• proof
• Proposition 3.2
• proof
• Definition 3.3
• Lemma 3.4
• proof
• proof : Proof of lower bound in Theorem \ref{['thm:main']}