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

Abstract

For $q \geq 2$, $n \in \mathbb{N}$, 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 $\left\{\exp\left(2\pi i\ell s_q(n)/b\right)\right\}_{n \in \mathbb{N}}$ and provide an explicit exponent in the upper bound.