  The Thue--Morse sequence $t=01101001\cdots$ is an automatic sequence over the
alphabet $\{0,1\}$. It can be defined as the binary sum-of-digits function
$s:\mathbb N\rightarrow\mathbb N$, reduced modulo $2$, or by using the
substitution $0\mapsto 01$, $1\mapsto 10$. We prove that the asymptotic density
of the set of natural numbers $n$ satisfying $t(n^3)=0$ 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 such 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 $\#\{n<N:t(p(n))=0\}$), or to sum-of-digits functions in
``sufficiently large bases''. By proving an asymptotic equivalence for the case
of the Thue--Morse sequence, and a cubic polynomial, we move one step closer to
the solution of the third Gelfond problem on the sum-of-digits function
(1967/1968), op. cit.
