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 $n$.

DOI: 10.5802/jtnb.63

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     author = {Patrick Gonzalez},
     title = {G\'en\'eralisation d'un r\'esultat de {Loxton} et van der {Poorten}},
     journal = {Journal de Th\'eorie des Nombres de Bordeaux},
     pages = {43--51},
     publisher = {Universit\'e Bordeaux I},
     volume = {4},
     number = {1},
     year = {1992},
     doi = {10.5802/jtnb.63},
     zbl = {0778.11015},
     mrnumber = {1183917},
     language = {fr},
     url = {https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.63/}
}

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Patrick Gonzalez. 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. doi : 10.5802/jtnb.63. https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.63/