Generalized additive bases, Konig's lemma, and the Erdos-Turan conjecture

Melvyn B. Nathanson

Published 2003-02-13, updated 2003-02-22Version 3

Let A be a set of nonnegative integers. For every nonnegative integer n and positive integer h, let r_{A}(n,h) 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 elements of A and a_1 \leq a_2 \leq ... \leq a_h. The infinite set A is called a basis of order h if r_{A}(n,h) \geq 1 for every nonnegative integer n. Erdos and Turan conjectured that limsup_{n\to\infty} r_A(n,2) = \infty for every basis A of order 2. This paper introduces a new class of additive bases and a general additive problem, a special case of which is the Erdos-Turan conjecture. Konig's lemma on the existence of infinite paths in certain graphs is used to prove that this general problem is equivalent to a related problem about finite sets of nonnegative integers.