Functiones et Approximatio Commentarii Mathematici

Representation functions of bases for binary linear forms

Melvyn B Nathanson

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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

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

First available in Project Euclid: 18 December 2008

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 11B34: Representation functions
Secondary: 11B13: Additive bases, including sumsets [See also 05B10] 11B75: Other combinatorial number theory 11A67: Other representations 11D04: Linear equations 11D72: Equations in many variables [See also 11P55]

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


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.

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