The level of distribution of the sum-of-digits function of linear recurrence number systems

Let $G=(G_j{)}_{j\ge 0}$ be a strictly increasing linear recurrent sequence of integers with $G_0=1$ having characteristic polynomial $X^d-a_1{X}^{d-1}-\cdots -a_{d-1}X-a_d$. It is well known that each positive integer $\nu$ can be uniquely represented by the so-called greedy expansion $\nu ={\epsilon }_0(\nu )G_0+\cdots +{\epsilon }_{\mathrm{\ell }}(\nu )G_{\mathrm{\ell }}$ for $\ell \in $\mathbb{N}$$satisfying$G_\ell \le \nu < G_{\ell+1}$. Here the digits are defined recursively in a way that $0\le \nu - \varepsilon_{\ell}(\nu) G_\ell - \cdots - \varepsilon_j(\nu) G_j < G_j$holdsfor$0 \le j \le \ell$. In the present paper we study the sum-of-digits function $s_G(\nu )={\epsilon }_0(\nu )+\cdots +{\epsilon }_{\mathrm{\ell }}(\nu )$ under certain natural assumptions on the sequence $G$. In particular, we determine its level of distribution $x^{\vartheta }$. To be more precise, we show that for $r,s\in \mathbb{N}$ with $\gcd (a_1+\cdots +a_d-1,s)=1$ we have for each $x\ge 1$ and all $A,\epsilon \in {\mathbb{R}}_{>0}$ that $$\sum _{q<x^{\vartheta -\epsilon }}\underset{z<x}{\max }\underset{1\le h\le q}{\max }\mid \sum _{\begin{array}{c}k<z,s_G(k)\equiv rs\\ k\equiv hq\end{array}}1-\frac{1}{q}\sum _{k<z,s_G(k)\equiv rs}1\mid \ll x(\log 2x{)}^{-A}\mathrm{.}$$ Here $\vartheta =\vartheta (G)\ge \frac{1}{2}$ can be computed explicitly and we have $\vartheta (G)\to 1$ for $a_1\to \mathrm{\infty }$. As an application we show that $#{ k\le x ,:, s_G(k) \equiv r \pmod{s}, ; k \hbox{ has at most two prime factors} } \gg x/\log x $ provided that the coefficient $a_1$ is not too small. Moreover, using Bombieri’s sieve an “almost prime number theorem” for $s_G$ follows from our result. Our work extends earlier results on the classical $q$-ary sum-of-digits function obtained by Fouvry and Mauduit.