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

Melvyn B. Nathanson^{[1]}

- [1] Department of Mathematics Lehman College (CUNY) Bronx, NY 10468 and CUNY Graduate Center New York, NY 10016

Journal de Théorie des Nombres de Bordeaux (2009)

- Volume: 21, Issue: 2, page 343-355
- ISSN: 1246-7405

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topNathanson, 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 -

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