## Functiones et Approximatio Commentarii Mathematici

### Representation functions of bases for binary linear forms

Melvyn B Nathanson

#### Abstract

Let $F(x_1,\ldots,x_m) = u_1 x_1 + \cdots + u_mx_m$ be a linear form with nonzero, relatively prime integer coefficients $u_1, \ldots, u_m$. For any set $A$ of integers, let $F(A)=\{F(a_1,\ldots,a_m): a_i \in A for i=1,\ldots,m\}.$ The {\it representation function} associated with the form $F$ is $$R_{A,F}(n) = \card ( \{ (a_1,\ldots,a_m)\in A^m: F(a_1,\ldots, a_m) = n \} ).$$ The set $A$ is a {\it basis with respect to $F$ for almost all integers} if the set ${\bf Z} \setminus F(A)$ has asymptotic density zero. Equivalently, the representation function of a basis for almost all integers is a function $f:{\bf Z} \rightarrow {\bf N_0}\cup\{\infty\}$ such that $f^{-1}(0)$ has density zero. Given such a function, the inverse problem for bases is to construct a set $A$ whose representation function is $f$. In this paper the inverse problem is solved for binary linear forms. for binary linear forms.

#### Article information

Source
Funct. Approx. Comment. Math., Volume 37, Number 2 (2007), 341-350.

Dates
First available in Project Euclid: 18 December 2008

https://projecteuclid.org/euclid.facm/1229619658

Digital Object Identifier
doi:10.7169/facm/1229619658

Mathematical Reviews number (MathSciNet)
MR2363831

Zentralblatt MATH identifier
1146.11007

#### Citation

Nathanson, Melvyn B. Representation functions of bases for binary linear forms. Funct. Approx. Comment. Math. 37 (2007), no. 2, 341--350. doi:10.7169/facm/1229619658. https://projecteuclid.org/euclid.facm/1229619658