# Representation functions of bases for binary linear forms

Melvyn B. Nathanson

Arxiv ID: 0709.0717•Last updated: 1/6/2021

Let F(x_1,...,x_m) = u_1 x_1 + ... + u_m x_m be a linear form with nonzero,
relatively prime integer coefficients u_1,..., u_m. For any set A of integers,
let F(A) = {F(a_1,...,a_m) : a_i in A for i=1,...,m}. The representation
function associated with the form F is
R_{A,F}(n) = card {(a_1,...,a_m) in A^m: F(a_1,..., a_m) = n}. The set A is a
basis with respect to F for almost all integers the set Z\F(A) has asymptotic
density zero. Equivalently, the representation function of an asymptotic basis
is a function f:Z -> N_0 U {\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.

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