The inverse problem for representation functions for general linear forms

The inverse problem for representation functions takes as input a triple (X,f,L), where X is a countable semigroup, f : X --> N_0 \cup {\infty} a function, L : a_1 x_1 + ... + a_h x_h an X-linear form and asks for a subset A \subseteq X such that there are f(x) solutions (counted appropriately) to L(x_1,...,x_h) = x for every x \in X, or a proof that no such subset exists. This paper represents the first systematic study of this problem for arbitrary linear forms when X = Z, the setting which in many respects is the most natural one. Having first settled on the "right" way to count representations, we prove that every primitive form has a unique representation basis, i.e.: a set A which represents the function f \equiv 1. We also prove that a partition regular form (i.e.: one for which no non-empty subset of the coefficients sums to zero) represents any function f for which {f^{-1}(0)} has zero asymptotic density. These two results answer questions recently posed by Nathanson. The inverse problem for partition irregular forms seems to be more complicated. The simplest example of such a form is x_1 - x_2, and for this form we provide some partial results. Several remaining open problems are discussed.