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 → N0 ∪ {∞} a function, L : a1x1 + · · ·+ ahxh an X-linear form and asks for a subset A ⊆ X such that there are f(x) solutions (counted appropriately) to L(x1, . . . , xh) = x for every x ∈ 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 ≡ 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 x1−x2, and for this form we provide some partial results. Several remaining open problems are discussed.