Let be a linear form with nonzero integer coefficients Let be an -tuple of finite sets of integers and let be an infinite set of integers. Define the representation function associated to the form and the sets and as follows :
If this representation function is constant, then the set is periodic and the period of will be bounded in terms of the diameter of the finite set Other results for complementing sets with respect to linear forms are also proved.
Soit une forme linéaire à coefficients entiers non nuls Soient un -uplet d’ensembles finis d’entiers et un ensemble infini d’entiers. Définissons la fonction de représentation associée à la forme et aux ensembles et comme suit :
Si cette fonction de représentation est constante, alors l’ensemble est périodique, et la période de est bornée en termes du diamètre de l’ensemble fini D’autres résultats sur les ensembles se complétant pour une forme linéaire sont également prouvés.
@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}, pages = {343--355}, publisher = {Universit\'e Bordeaux 1}, volume = {21}, number = {2}, year = {2009}, doi = {10.5802/jtnb.675}, mrnumber = {2541430}, language = {en}, url = {http://archive.numdam.org/articles/10.5802/jtnb.675/} }
TY - JOUR AU - Nathanson, Melvyn B. TI - Problems in additive number theory, II: Linear forms and complementing sets JO - Journal de théorie des nombres de Bordeaux PY - 2009 SP - 343 EP - 355 VL - 21 IS - 2 PB - Université Bordeaux 1 UR - http://archive.numdam.org/articles/10.5802/jtnb.675/ DO - 10.5802/jtnb.675 LA - en ID - JTNB_2009__21_2_343_0 ER -
%0 Journal Article %A Nathanson, Melvyn B. %T Problems in additive number theory, II: Linear forms and complementing sets %J Journal de théorie des nombres de Bordeaux %D 2009 %P 343-355 %V 21 %N 2 %I Université Bordeaux 1 %U http://archive.numdam.org/articles/10.5802/jtnb.675/ %R 10.5802/jtnb.675 %G en %F 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://archive.numdam.org/articles/10.5802/jtnb.675/
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