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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} afunction, 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) toL(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 forarbitrary linear forms when X = Z, the setting which in many respects is themost natural one. Having first settled on the "right" way to countrepresentations, we prove that every primitive form has a unique representationbasis, i.e.: a set A which represents the function f \equiv 1. We also provethat a partition regular form (i.e.: one for which no non-empty subset of thecoefficients sums to zero) represents any function f for which {f^{-1}(0)} haszero asymptotic density. These two results answer questions recently posed byNathanson.  The inverse problem for partition irregular forms seems to be morecomplicated. The simplest example of such a form is x_1 - x_2, and for thisform we provide some partial results. Several remaining open problems arediscussed.

The inverse problem for representation functions for general linear forms