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

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 ${\sum }_{n⩽x}|\hat{I}\cdot (n)|=o(x^{\hat{I}}\pm )$ for any $\hat{I}\pm >\log (3)/\log (4)$. In this paper, we sharpen this result, providing upper and lower bounds for $\hat{I}\pm$. 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.

9 pages, 4 figures, accepted to Experimental Mathematics