Thue--Morse along the sequence of cubes

Abstract

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.