The level of distribution of the sum-of-digits function in arithmetic progressions

For $q\ge 2$, $n\in \mathbb{N}$, let $s_q(n)$ denote the sum of the digits of $n$ written in base $q$. Spiegelhofer (2020) proved that the Thue–Morse sequence has level of distribution $1$, improving on a former result of Fouvry and Mauduit (1996). In this paper we generalize this result to sequences of type ${\{\exp (2\pi i\mathrm{\ell }{s}_q(n)\mathrm{/}b)\}}_{n\in \mathbb{N}}$ and provide an explicit exponent in the upper bound.