Representation Functions of Bases for Binary Linear Forms

Let F (x1, . . . , xm) = u1x1 + · · · + umxm be a linear form with relatively prime integer coefficients u1, . . . , um. For any set A of integers, let F (A) = {F (a1, . . . , am) : ai ∈ A for i = 1, . . . , m}. The representation function associated with the form F is RA,F (n) = card ({(a1, . . . , am) ∈ A m : F (a1, . . . , am) = n}) . The set A is an asymptotic basis with respect to F if RA,F (n) ≥ 1 for all but finitely many integers n. Thus, the representation function of an asymptotic basis is a function f : Z → N0 ∪ {∞} such that f(0) is finite. Given such a function, the inverse problem for asymptotic bases is to construct a set A whose representation function is f . In this paper the inverse problem is solved for binary linear forms. 1. Direct and inverse problems for representation functions Let N, N0, and Z denote the sets of positive integers, nonnegative integers, and integers, respectively. In additive number theory, a classical direct problem is to describe the integers that can be represented as the sum of a bounded number of elements of a fixed set A of integers. For example, describe the integers that are sums of two primes or three squares or four cubes. Given a set A and a positive integer m, we associate various representation functions to the sumset mA = {a1 + · · · + am : ai ∈ A for i = 1, . . . , m}. The two most important are the ordered representation function RA,m(n) = {(a1, . . . , am) ∈ A m : m