Representation functions of bases for binary linear forms

TL;DR

This paper solves the inverse problem for bases with respect to binary linear forms, showing how to construct sets of integers with prescribed representation functions.

Contribution

It provides a solution to the inverse problem for bases related to binary linear forms, a problem previously unresolved.

Findings

Constructed sets with prescribed representation functions

Proved that almost all integers are represented by such bases

Established asymptotic density properties of these sets

Abstract

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.