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

Abstract

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 greedy expansion $\nu=\varepsilon_0(\nu)G_0+\cdots+\varepsilon_\ell(\nu)G_\ell$ for $\ell \in \mathbb{N}$ satisfying $G_\ell \le \nu < G_{\ell+1}$. Here the digits are defined recursively in a way that $0\le \nu - \varepsilon_{\ell}(\nu) G_\ell - \cdots - \varepsilon_j(\nu) G_j < G_j$ holds for $0 \le j \le \ell$. In the present paper we study the 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 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 $A,\varepsilon\in\mathbb{R}_{>0}$ that \[ \sum_{q<x^{\vartheta-\varepsilon}}\max_{z<x}\max_{1\leq h\leq q} \lvert\sum_{\substack{k<z,s_G(k)\equiv r\bmod s\\ k\equiv h\bmod q}}1 -\frac1q\sum_{k<z,s_G(k)\equiv r\bmod s}1\rvert \ll x(\log 2x)^{-A}. \] 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 \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.