The inverse problem for representation functions for general linear forms

TL;DR

This paper systematically studies the inverse problem for representation functions of linear forms over integers, proving existence and uniqueness results for primitive and partition regular forms, and discussing open problems for irregular forms.

Contribution

It introduces a systematic approach to the inverse problem for general linear forms over integers, establishing unique bases and representation conditions for primitive and partition regular forms.

Findings

Primitive forms have unique representation bases.

Partition regular forms represent functions with zero-density zero sets.

Partial results are provided for partition irregular forms.

Abstract

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.