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

TL;DR

This paper refines bounds on the growth of the absolute autocorrelations of the Thue-Morse sequence, providing sharper estimates and insights into the structure of related regular sequences.

Contribution

It improves existing bounds on the sum of absolute autocorrelations of the Thue-Morse sequence and explores the structure of the product of regular sequences.

Findings

Established tighter bounds for autocorrelation sums.

Derived the structure of the linear representation of product sequences.

Enhanced understanding of regular sequence interactions.

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 $\sum_{n\leqslant x}|\eta(n)|=o(x^\alpha)$ for any $\alpha>\log(3)/\log(4)$. In this paper, we sharpen this result, providing upper and lower bounds for $\alpha$. 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.