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

• [1] Department of Mathematics Lehman College (CUNY) Bronx, NY 10468 and CUNY Graduate Center New York, NY 10016
• Volume: 21, Issue: 2, page 343-355
• ISSN: 1246-7405

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## Abstract

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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.

## How to cite

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Nathanson, Melvyn B.. "Problems in additive number theory, II: Linear forms and complementing sets." Journal de Théorie des Nombres de Bordeaux 21.2 (2009): 343-355. <http://eudml.org/doc/10885>.

@article{Nathanson2009,
abstract = {Let $\varphi (x_1,\ldots ,x_h,y) = u_1x_1 + \cdots + u_hx_h+vy$ be a linear form with nonzero integer coefficients $u_1,\ldots , u_h, v.$ Let $\mathcal\{A\} = (A_1,\ldots , A_h)$ 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 $\mathcal\{A\}$ and $B$ as follows :R^\{(\varphi )\}\_\{ \mathcal\{A\} ,B\}(n) = \text\{card\}\left( \begin\{aligned\} \Bigl \lbrace (a\_1,\ldots , a\_h,b) \in A\_1 &\times \cdots \times A\_h \times B:\\ & \varphi (a\_1, \ldots , a\_h,b ) = n \Bigr \rbrace \end\{aligned\} \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 $\lbrace \varphi (a_1,\ldots ,a_h,0): (a_1,\ldots , a_h) \in A_1 \times \cdots \times A_h\rbrace .$ Other results for complementing sets with respect to linear forms are also proved.},
affiliation = {Department of Mathematics Lehman College (CUNY) Bronx, NY 10468 and CUNY Graduate Center New York, NY 10016},
author = {Nathanson, Melvyn B.},
journal = {Journal de Théorie des Nombres de Bordeaux},
keywords = {Representation functions; linear forms; complementing sets; tiling by finite sets; inverse problems in additive number theory; representation by linear forms; -complementing pairs},
language = {eng},
number = {2},
pages = {343-355},
publisher = {Université Bordeaux 1},
title = {Problems in additive number theory, II: Linear forms and complementing sets},
url = {http://eudml.org/doc/10885},
volume = {21},
year = {2009},
}

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
PB - Université Bordeaux 1
VL - 21
IS - 2
SP - 343
EP - 355
AB - Let $\varphi (x_1,\ldots ,x_h,y) = u_1x_1 + \cdots + u_hx_h+vy$ be a linear form with nonzero integer coefficients $u_1,\ldots , u_h, v.$ Let $\mathcal{A} = (A_1,\ldots , A_h)$ 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 $\mathcal{A}$ and $B$ as follows :R^{(\varphi )}_{ \mathcal{A} ,B}(n) = \text{card}\left( \begin{aligned} \Bigl \lbrace (a_1,\ldots , a_h,b) \in A_1 &\times \cdots \times A_h \times B:\\ & \varphi (a_1, \ldots , a_h,b ) = n \Bigr \rbrace \end{aligned} \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 $\lbrace \varphi (a_1,\ldots ,a_h,0): (a_1,\ldots , a_h) \in A_1 \times \cdots \times A_h\rbrace .$ Other results for complementing sets with respect to linear forms are also proved.
LA - eng
KW - Representation functions; linear forms; complementing sets; tiling by finite sets; inverse problems in additive number theory; representation by linear forms; -complementing pairs
UR - http://eudml.org/doc/10885
ER -

## References

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6. —, Generalized additive bases, König’s lemma, and the Erdős-Turán conjecture. J. Number Theory 106 (2004), no. 1, 70–78. Zbl1090.11010MR2049593
7. Donald J. Newman, Tesselation of integers. J. Number Theory 9 (1977), no. 1, 107–111. Zbl0348.10038MR429720
8. Ivan Niven, A characterization of complementing sets of pairs of integers. Duke Math. J. 38 (1971), 193–203. Zbl0214.30503MR274414
9. John P. Steinberger, Tilings of the integers can have superpolynomial periods. Preprint, 2005. Zbl1199.11061
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. Zbl1103.68103MR2223402

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