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
$X^{d}-a_1X^{d-1}-\cdots-a_{d-1}X-a_d$. It is well known that each
positive integer $\nu$ can be uniquely represented by the so-called
\emph{greedy expansion}
$\nu=\varepsilon_0(\nu)G_0+\cdots+\varepsilon_\ell(\nu)G_\ell$ for
$\ell \in \N$ satisfying $G_\ell \le \nu < G_{\ell+1}$. Here the
\emph{digits} are defined recursively in a way that
$0\le \nu - \varepsilon_{\ell}(\nu) G_\ell - \cdots -
\varepsilon_j(\nu) G_j < G_j$ holds for $0 \le j \le \ell$. In the
present paper we study the \emph{sum-of-digits function}
$s_G(\nu)=\varepsilon_0(\nu)+\cdots+\varepsilon_\ell(\nu)$ under
certain natural assumptions on the sequence $G$. In particular, we
determine its \emph{level of distribution} $x^{\vartheta}$. To be
more precise, we show that for $r,s\in\N$ with
$\gcd(a_1+\cdots+a_d-1,s)=1$ we have for each $x\ge 1$ and all
$A,\varepsilon\in\R_{>0}$ that
\begin{multline*}
\sum_{q<x^{\vartheta-\varepsilon}}\max_{z<x}\max_{1\leq h\leq q}
\lvert\sum_{\substack{k<z,s_G(k)\equiv r\bmod s\\ k\equiv h\bmod
q}}1 -\frac1q\sum_{k<z,s_G(k)\equiv r\bmod s}1\rvert\\ \ll
x(\log 2x)^{-A}.
\end{multline*}
Here $\vartheta=\vartheta(G) \ge \frac12$ can be computed explicitly
and we have $\vartheta(G) \to 1$ for $a_1\to\infty$. As an
application we show that
$\#\{ k\le x \,:\, s_G(k) \equiv r \pmod{s}, \; k$ 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.