Problems in additive number theory. II: Linear forms and complementing sets
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 finite sets of integers and let \(B\) an infinite set of integers. If the pair \(({\mathcal A},B)\) is \(t\)-complementing with respect to \(\varphi\), then \(B\) is periodic; that is, there is an integer \(m\in\mathbb N\) such that \(B\) is a union of congruence classes modulo \(m\). In the special case \(h= t= 1\), \(\psi(x)= x\) and \(\nu= 1\) one obtains \(\varphi(x, y)= x+ y\) and the theorem above gives a result of \textit{D. J. Newman} [J. Number Theory 9, 107--111 (1977; Zbl 0348.10038)]. In this paper the author also proves further results for complementing sets with respect to linear forms. For Part I, see CRM Proc. Lect. Notes 43, 263--270 (2007; Zbl 1183.11006); Part III has been published in Advanced Courses in Mathematics - CRM Barcelona, 279--297 (2009; Zbl 1221.11023).
scientific article; zbMATH DE number 5620655