The level of distribution of the sum-of-digits function in arithmetic progressions
Nathan Toumi
TL;DR
This work extends Spiegelhofer’s level of distribution results for the Thue–Morse sequence to the base-$q$ generalization $t_q(n)=\mathrm{e}\left(\dfrac{\ell}{b}s_q(n)\right)$, providing an explicit exponent $\eta$ in averaged distribution bounds over arithmetic progressions. The authors develop a base-$q$ carry-propagation framework, a recurrence on a finite-state graph, and a sequence of digit-cutting and Farey-approximation steps, all fed into iterative van der Corput reductions that reduce exponential sums to controlled Gowers-norm-type quantities. They prove an explicit, effective contraction exponent $\eta_0$ for the transfer-graph, then combine this with iterative digit-shifting to obtain a bound of the form $\sum_{D<m<qD}\max... \leq Cx^{1-\eta}$ with an explicit $\eta=\eta(\varepsilon,b,q)$, including a concrete value for the $q=b=2$ case. The results yield a Bombieri–Vinogradov-type average for $s_q(n)$ in progressions with an explicit exponent, advancing the understanding of digit-sum distribution in arithmetic progressions beyond the binary Thue–Morse setting and enabling quantitative conclusions for explicit bases and moduli.
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 $\left\{\exp\left(2πi\ell s_q(n)/b\right)\right\}_{n \in \mathbb{N}}$ and provide an explicit exponent in the upper bound.