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 $\varphi \left({x}_{1},...,{x}_{h},y\right)={u}_{1}{x}_{1}+\cdots +{u}_{h}{x}_{h}+vy$ 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 : 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/

[1] András Biró, Divisibility of integer polynomials and tilings of the integers. Acta Arith. 118 (2005), no. 2, 117–127. | MR 2141045 | Zbl 1088.11016

[2] Rodney T. Hansen, Complementing pairs of subsets of the plane. Duke Math. J. 36 (1969), 441–449. | MR 244404 | Zbl 0181.05304

[3] Mihail N. Kolountzakis, Translational tilings of the integers with long periods. Electron. J. Combin. 10 (2003), Research Paper 22, 9 pp. (electronic). | MR 1975772 | Zbl 1107.11016

[4] Jeffrey C. Lagarias, Yang Wang, Tiling the line with translates of one tile? Invent. Math. 124 (1996), no. 1-3, 341–365. | MR 1369421 | Zbl 0847.05037

[5] Melvyn B. Nathanson, Complementing sets of $n$-tuples of integers. Proc. Amer. Math. Soc. 34 (1972), 71–72. | MR 294286 | Zbl 0249.10049

[6] —, Generalized additive bases, König’s lemma, and the Erdős-Turán conjecture. J. Number Theory 106 (2004), no. 1, 70–78. | MR 2049593 | Zbl 1090.11010

[7] Donald J. Newman, Tesselation of integers. J. Number Theory 9 (1977), no. 1, 107–111. | MR 429720 | Zbl 0348.10038

[8] Ivan Niven, A characterization of complementing sets of pairs of integers. Duke Math. J. 38 (1971), 193–203. | MR 274414 | Zbl 0214.30503

[9] John P. Steinberger, Tilings of the integers can have superpolynomial periods. Preprint, 2005.

[10] Mario Szegedy, Algorithms to tile the infinite grid with finite clusters. Preprint available on www.cs.rutgers.edu/ szegedy/, 1998.

[11] Robert Tijdeman, Periodicity and almost-periodicity. More sets, graphs and numbers, Bolyai Soc. Math. Stud., vol. 15, Springer, Berlin, 2006, pp. 381–405. | MR 2223402 | Zbl 1103.68103