Problems in additive number theory. II: Linear forms and complementing sets
type: Item
instance of: scholarly article
title: Problems in additive number theory. II: Linear forms and complementing sets
zbMATH Open document ID: 1239.11025
author: Melvyn B. Nathanson
published in: Journal de Théorie des Nombres de Bordeaux
publication date: 2009-10-26T00:00:00Z
full work available at URL: https://eudml.org/doc/10885
review text: Let \(S(A, B):=\{a+b: a\in A,\, b\in B\}\) denote the sumset of the sets \(A\) and \(B\) of integers. The pair \((A, B)\) is called a complementing pair if every element of the sumset \(S(A,B)\) has a unique representation as the sum of an element of \(A\) and an element of \(B\); that is if \(a_1+b_1= a_2+ b_2\) (\(a_1,a_2\in A\), \(b_1,b_2\in B\)) then \(a_1= a_2\) and \(b_1= b_2\). The problem of complementing pairs for the set of integers is a special case of the general problem of the representation of integers by linear forms. Let \(\psi(x_1,\dots, x_h)= u_1x_1+\cdots+ u_h x_h\) be a linear form and \({\mathcal A}= (A_1,\dots, A_h)\) be an \(h\)-tuple of sets of integers. The image of \(\psi\) with respect to \({\mathcal A}\) is the set \[ \psi({\mathcal A}= \{\psi(a_1,\dots, a_h): (a_1,\dots, a_h)\in A_1\times\cdots\times A_h\}. \] With \(\omega(y)= \nu y\) one considers the linear form \(\varphi(x_1,\dots, x_h)= \psi(x_1,\dots, x_h)+ \omega(y)\). For an \(h\)-tuple \({\mathcal A}= (A_1,\dots, A_h)\) of sets of integers and a set \(B\) of integers is the image of \(\varphi\) with respect to \(({\mathcal A}, B)\) the set \(\varphi({\mathcal A},B)= \psi({\mathcal A})+\omega(B)\). The representation function associated to \(\varphi\), \({\mathcal A}\) and B is defined as \[ R(n)= \text{card}(\{(a_1,\dots, a_h,b)\in A_1\times\cdots\times A_h\times B: \varphi(a_1,\dots, a_h, b)= n\}). \] For an \(h\)-tuple \({\mathcal A}\) of sets of integers and a set \(B\) of integers the pair \(({\mathcal A},B)\) is called \(t\)-complementing with respect to the linear form \[ \varphi(x_1,\dots, x_h,y)= u_1 x_1+\cdots+ u_h x_h+\nu y\quad\text{if }R(n)= t\quad\text{for all }n. \] As a result of this note the author presents Theorem 1: Let \(\psi(x_1,\dots, x_h)= u_1 x_1+\cdots+ u_h x_h\) and \(\varphi(x_1,\dots, x_h, y)= \psi(x_1,\dots, x_h)+\nu y\) with integer coefficients \(u_1,\dots, u_h\), \(\nu\neq 0\) Let \({\mathcal A}= \{A_1,\dots, A_h\}\) be an \(h\)-tuple of nonempty …
reviewed by: Erich Härtter
Mathematics Subject Classification ID: 11B75
Mathematics Subject Classification ID: 11B34
Mathematics Subject Classification ID: 11P70
zbMATH DE Number: 5620655
zbMATH Keywords: complementing sets
zbMATH Keywords: representation by linear forms
zbMATH Keywords: \(t\)-complementing pairs
MaRDI profile type: Publication
OpenAlex ID: W2315595928
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DOI: 10.5802/JTNB.675
description: scientific article; zbMATH DE number 5620655