Généralisation d'un résultat de Loxton et van der Poorten
Journal de théorie des nombres de Bordeaux, Volume 4 (1992) no. 1, p. 43-51

In this paper we propose to generalise a result of Loxton and Van der Poorten [1] in the following way. Let $g$ be an integer greater than or equal to two and $C$ be a finite subset of $ℕ$ containing zero. $C$ will be said to be $g$-free if, for all integers $n$, the equality ${\sum }_{i=0}^{n}{c}_{i}{g}^{i}={\sum }_{i=0}^{n}{c}_{i}^{\text{'}}{g}^{i}$with ${c}_{i},{c}_{i}^{\text{'}}\in C,0\le i\le n$for$i=0,\cdots ,n$Let $P$be the polynomial ${\sum }_{c\in C}{X}^{c},{P}_{n}={\prod }_{i=0}^{n-1}P\left({X}^{g}h\right)$and ${U}_{g}n$ the set of the ${g}^{n}$-th roots of unity. We shall prove that $C$is$g$-free, when $\mathrm{Card}C=g$, if and only if, $\mathrm{Card}\left\{x\in {U}_{{g}^{n}},{P}_{n}\left(x\right)\ne 0\right\}$ is bounded independently of $n$.

@article{JTNB_1992__4_1_43_0,
author = {Gonzalez, Patrick},
title = {G\'en\'eralisation d'un r\'esultat de Loxton et van der Poorten},
journal = {Journal de th\'eorie des nombres de Bordeaux},
publisher = {Universit\'e Bordeaux I},
volume = {4},
number = {1},
year = {1992},
pages = {43-51},
zbl = {0778.11015},
mrnumber = {1183917},
language = {fr},
url = {http://www.numdam.org/item/JTNB_1992__4_1_43_0}
}

Gonzalez, Patrick. Généralisation d'un résultat de Loxton et van der Poorten. Journal de théorie des nombres de Bordeaux, Volume 4 (1992) no. 1, pp. 43-51. http://www.numdam.org/item/JTNB_1992__4_1_43_0/

[1] J.H. Loxton et A. J. Van Der Poorten, An awful problem about integers in base 4, Acta Arith. 49 (1987), 193-203. | MR 928637 | Zbl 0636.10003

[2] G. Rauzy, Systèmes de numération, Journées de théorie élémentaire et analytique des nombres, 1982, Valenciennes.