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

Abstract

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Let ϕ ( x 1 , ... , x h , y ) = u 1 x 1 + ⋯ + u h x h + v y 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 ϕ and the sets 𝒜 and B as follows : R 𝒜 , B ( ϕ ) ( n ) = card { ( a 1 , ... , a h , b ) ∈ A 1 × ⋯ × A h × B : ϕ ( a 1 , ... , a h , b ) = n } . 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 { ϕ ( a 1 , ... , a h , 0 ) : ( a 1 , ... , a h ) ∈ A 1 × ⋯ × A h } . 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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  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
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