Généralisation d'un résultat de Loxton et van der Poorten

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 $\mathbb{N}$ 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 $n$.

[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.