Problems in additive number theory, II: Linear forms and complementing sets

Nathanson, Melvyn B.

Journal de théorie des nombres de Bordeaux, Tome 21 (2009) no. 2, pp. 343-355.

Soit $\varphi (x_1,...,x_h,y)=u_1{x}_1+\cdots +u_h{x}_h+vy$ 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 $\varphi$ et aux ensembles $𝒜$ et $B$ comme suit :

$$R_{𝒜,B}^{\left(\varphi \right)}\left(n\right)=\text{card}\left(\begin{array}{cc}\hfill \{(a_1,...,a_h,b)\in A_1& \times \cdots \times A_h\times B:\hfill \\ & \varphi (a_1,...,a_h,b)=n\}\hfill \end{array}\right).$$

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 $\{\varphi (a_1,...,a_h,0):(a_1,...,a_h)\in A_1\times \cdots \times A_h\}.$ D’autres résultats sur les ensembles se complétant pour une forme linéaire sont également prouvés.

Let $\varphi (x_1,...,x_h,y)=u_1{x}_1+\cdots +u_h{x}_h+vy$ 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 $\varphi$ and the sets $𝒜$ and $B$ as follows :

$$R_{𝒜,B}^{\left(\varphi \right)}\left(n\right)=\text{card}\left(\begin{array}{cc}\hfill \{(a_1,...,a_h,b)\in A_1& \times \cdots \times A_h\times B:\hfill \\ & \varphi (a_1,...,a_h,b)=n\}\hfill \end{array}\right).$$

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 $\{\varphi (a_1,...,a_h,0):(a_1,...,a_h)\in A_1\times \cdots \times A_h\}.$ Other results for complementing sets with respect to linear forms are also proved.

Mots clés : Representation functions, linear forms, complementing sets, tiling by finite sets, inverse problems in additive number theory.

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