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