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

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

2003-02-10

arxiv.org/abs/math/0302091v2

Mathematics

Number Theory

15 pages. LaTex file. Corrected and revised manuscript

Scientific paper

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.

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