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September 2007 Representation functions of bases for binary linear forms
Melvyn B Nathanson
Funct. Approx. Comment. Math. 37(2): 341-350 (September 2007). DOI: 10.7169/facm/1229619658

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.

Citation

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Melvyn B Nathanson. "Representation functions of bases for binary linear forms." Funct. Approx. Comment. Math. 37 (2) 341 - 350, September 2007. https://doi.org/10.7169/facm/1229619658

Information

Published: September 2007
First available in Project Euclid: 18 December 2008

zbMATH: 1146.11007
MathSciNet: MR2363831
Digital Object Identifier: 10.7169/facm/1229619658

Subjects:
Primary: 11B34
Secondary: 11A67 , 11B13 , 11B75 , 11D04 , 11D72

Keywords: $B_h[g]$ and $B_F[g]$ sets , additive bases , Erdős-Turán conjecture , linear forms , representation functions , Sidon sets

Rights: Copyright © 2007 Adam Mickiewicz University

Vol.37 • No. 2 • September 2007
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