Every function is the representation function of an additive basis for the integers

TL;DR

This paper proves that any function with finitely many zeros can be realized as the representation function of an additive basis of integers, and such bases can be made arbitrarily sparse.

Contribution

It establishes that every suitable function can be represented as a sum representation function of an integer set, including very sparse sets.

Findings

Any function with finitely many zeros can be realized as a representation function.

Such sets can be constructed to be arbitrarily sparse.

The result applies to all integers, with only finitely many exceptions.

Abstract

Let A be a set of integers. For every integer n, let r_{A,h}(n) denote the number of representations of n in the form n = a_1 + a_2 + ... + a_h, where a_1, a_2,...,a_h are in A and a_1 \leq a_2 \leq ... \leq a_h. The function r_{A,h}: Z \to N_0 \cup \infty is the representation function of order h for A. The set A is called an asymptotic basis of order h if r_{A,h}^{-1}(0) is finite, that is, if every integer with at most a finite number of exceptions can be represented as the sum of exactly h not necessarily distinct elements of A. It is proved that every function is a representation function, that is, if f: Z \to N_0 \cup \infty is any function such that f^{-1}(0) is finite, then there exists a set A of integers such that f(n) = r_{A,h}(n) for all n in Z. Moreover, the set A can be arbitrarily sparse in the sense that, if \phi(x) \to \infty, then there exists a set A with f(n) = r_{A,h}(n) such that card{a in A : |a| \leq x} < \phi(x) for all sufficiently large x.