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

Abstract

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

57 pages, 3 figures