Problems in additive number theory, II: Linear forms and complementing sets
Journal de théorie des nombres de Bordeaux, Volume 21 (2009) no. 2, p. 343-355

Let ϕ(x 1 ,...,x h ,y)=u 1 x 1 ++u h x h +vy be a linear form with nonzero integer coefficients u 1 ,...,u h ,v. Let 𝒜=(A 1 ,...,A h ) be an h-tuple of finite sets of integers and let B be an infinite set of integers. Define the representation function associated to the form ϕ and the sets 𝒜 and B as follows :

R𝒜,B(ϕ)(n)=card{(a1,...,ah,b)A1××Ah×B:ϕ(a1,...,ah,b)=n}.

If this representation function is constant, then the set B is periodic and the period of B will be bounded in terms of the diameter of the finite set {ϕ(a 1 ,...,a h ,0):(a 1 ,...,a h )A 1 ××A h }. Other results for complementing sets with respect to linear forms are also proved.

Soit ϕ(x 1 ,...,x h ,y)=u 1 x 1 ++u h x h +vy une forme linéaire à coefficients entiers non nuls u 1 ,...,u h ,v. Soient 𝒜=(A 1 ,...,A h ) un h-uplet d’ensembles finis d’entiers et B un ensemble infini d’entiers. Définissons la fonction de représentation associée à la forme ϕ et aux ensembles 𝒜 et B comme suit :

R𝒜,B(ϕ)(n)=card{(a1,...,ah,b)A1××Ah×B:ϕ(a1,...,ah,b)=n}.

Si cette fonction de représentation est constante, alors l’ensemble B est périodique, et la période de B est bornée en termes du diamètre de l’ensemble fini {ϕ(a 1 ,...,a h ,0):(a 1 ,...,a h )A 1 ××A h }. D’autres résultats sur les ensembles se complétant pour une forme linéaire sont également prouvés.

DOI : https://doi.org/10.5802/jtnb.675
Keywords: Representation functions, linear forms, complementing sets, tiling by finite sets, inverse problems in additive number theory.
@article{JTNB_2009__21_2_343_0,
     author = {Nathanson, Melvyn B.},
     title = {Problems in additive number theory, II: Linear forms and complementing sets},
     journal = {Journal de th\'eorie des nombres de Bordeaux},
     publisher = {Universit\'e Bordeaux 1},
     volume = {21},
     number = {2},
     year = {2009},
     pages = {343-355},
     doi = {10.5802/jtnb.675},
     mrnumber = {2541430},
     zbl = {pre05620655},
     language = {en},
     url = {http://www.numdam.org/item/JTNB_2009__21_2_343_0}
}
Nathanson, Melvyn B. Problems in additive number theory, II: Linear forms and complementing sets. Journal de théorie des nombres de Bordeaux, Volume 21 (2009) no. 2, pp. 343-355. doi : 10.5802/jtnb.675. http://www.numdam.org/item/JTNB_2009__21_2_343_0/

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