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- W4307169596 abstract "Let $G=(G_j)_{jge 0}$ 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 $0le nu - varepsilon_{ell}(nu) G_ell - cdots - varepsilon_j(nu) G_j < G_j$ 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 $r,sinmathbb{N}$ with $gcd(a_1+cdots+a_d-1,s)=1$ we have for each $xge 1$ and all $A,varepsiloninmathbb{R}_{>0}$ that [ sum_{q<x^{vartheta-varepsilon}}max_{z<x}max_{1leq hleq q} lvertsum_{substack{k<z,s_G(k)equiv rbmod s kequiv hbmod q}}1 -frac1qsum_{k<z,s_G(k)equiv rbmod s}1rvert ll x(log 2x)^{-A}. ] Here $vartheta=vartheta(G) ge frac12$ can be computed explicitly and we have $vartheta(G) to 1$ for $a_1toinfty$. As an application we show that $#{ kle 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." @default.
- W4307169596 created "2022-10-29" @default.
- W4307169596 creator A5020243260 @default.
- W4307169596 creator A5045161708 @default.
- W4307169596 date "2019-09-18" @default.
- W4307169596 modified "2023-09-28" @default.
- W4307169596 title "The level of distribution of the sum-of-digits function of linear recurrence number systems" @default.
- W4307169596 doi "https://doi.org/10.48550/arxiv.1909.08499" @default.
- W4307169596 hasPublicationYear "2019" @default.
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