The inverse problem for representation functions of additive bases

Let A be a set of integers. For every integer n, let r_{A,2}(n) denote the number of representations of n in the form n = a_1 + a_2, where a_1 and a_2 are in A and a_1 \leq a_2. The function r_{A,2}: Z \to N_0 \cup {\infty} is the representation function of order 2 for A. The set A is called an asymptotic basis of order 2 if r_{A,2}^{-1}(0) is finite, that is, if every integer with at most a finite number of exceptions can be represented as the sum of two 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,2}(n) for all n \in \Z. Moreover, the set A can be constructed so that card{a\in A : |a| \leq x} \gg x^{1/3}.