Thue--Morse along the sequence of cubes

Let $\mathsf t=\mathtt{01101001}\cdots$ be the Thue--Morse word over the alphabet $\{\mathtt 0,\mathtt 1\}$. This automatic sequence can be defined as the binary sum-of-digits function $\mathsf s:\mathbb N\rightarrow\mathbb N$, reduced modulo $2$. We prove that the asymptotic density of the set of natural numbers $n$ satisfying $\mathsf t(n^3)=\mathtt 0$ equals $1/2$. Comparable results, featuring asymptotic equivalence along a polynomial as in our theorem, were previously only known for the linear case [A. O. Gelfond, Acta Arith. 13 (1967/68), 259--265], and for the sequence of squares. The main theorem in [C. Mauduit and J. Rivat, Acta Math. 203 (2009), no. 1, 107--148] was the first result for the sequence of squares. Concerning the sum-of-digits function along polynomials $p$ of degree at least three, previous results were restricted either to lower bounds (such as for the numbers $\#\{n更多