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.

Let $\varphi \left({x}_{1},...,{x}_{h},y\right)={u}_{1}{x}_{1}+\cdots +{u}_{h}{x}_{h}+vy$ be a linear form with nonzero integer coefficients ${u}_{1},...,{u}_{h},v.$ Let $𝒜=\left({A}_{1},...,{A}_{h}\right)$ 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 :

 ${R}_{𝒜,B}^{\left(\varphi \right)}\left(n\right)=\text{card}\left(\begin{array}{cc}\hfill \left\{\left({a}_{1},...,{a}_{h},b\right)\in {A}_{1}& ×\cdots ×{A}_{h}×B:\hfill \\ & \varphi \left({a}_{1},...,{a}_{h},b\right)=n\right\}\hfill \end{array}\right).$

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 $\left\{\varphi \left({a}_{1},...,{a}_{h},0\right):\left({a}_{1},...,{a}_{h}\right)\in {A}_{1}×\cdots ×{A}_{h}\right\}.$ Other results for complementing sets with respect to linear forms are also proved.

Soit $\varphi \left({x}_{1},...,{x}_{h},y\right)={u}_{1}{x}_{1}+\cdots +{u}_{h}{x}_{h}+vy$ une forme linéaire à coefficients entiers non nuls ${u}_{1},...,{u}_{h},v.$ Soient $𝒜=\left({A}_{1},...,{A}_{h}\right)$ 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 :

 ${R}_{𝒜,B}^{\left(\varphi \right)}\left(n\right)=\text{card}\left(\begin{array}{cc}\hfill \left\{\left({a}_{1},...,{a}_{h},b\right)\in {A}_{1}& ×\cdots ×{A}_{h}×B:\hfill \\ & \varphi \left({a}_{1},...,{a}_{h},b\right)=n\right\}\hfill \end{array}\right).$

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 $\left\{\varphi \left({a}_{1},...,{a}_{h},0\right):\left({a}_{1},...,{a}_{h}\right)\in {A}_{1}×\cdots ×{A}_{h}\right\}.$ D’autres résultats sur les ensembles se complétant pour une forme linéaire sont également prouvés.

DOI: 10.5802/jtnb.675
Keywords: Representation functions, linear forms, complementing sets, tiling by finite sets, inverse problems in additive number theory.
Melvyn B. Nathanson 1

1 Department of Mathematics Lehman College (CUNY) Bronx, NY 10468 and CUNY Graduate Center New York, NY 10016
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Melvyn B. Nathanson. 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. https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.675/

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