Results for the behaviour of automatic sequences along polynomials of degree higher than two are sparse. We evaluate the Thue--Morse sequence 01101001... along the sequence of cubes, and prove that both values are taken with asymptotic frequency equal to 1/2.
Concerning polynomials of degree at least 3, lower bounds existed before, as well as results for the sum-of-digits function in "sufficiently large bases" q (depending on the polynomial). This result extends Mauduit and Rivat's breakthrough results on the sum of digits of squares and primes, and answers another case of the last Gelfond problem on the sum-of-digits function.