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 containing zero. C will be said to be g-free if, for all integers n, the equality i = 0 n c i g i = i = 0 n c i ' g i with c i , c i ' C , 0 i n for i = 0 , , n Let P be the polynomial c C X c , P n = i = 0 n - 1 P ( X g h ) and U g n the set of the g n -th roots of unity. We shall prove that C is g -free, when Card C = g , if and only if, Card x U g n , P n ( x ) 0 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.