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

JOURNAL OF NUMBER THEORY

卷:106

Article

Nathanson, MB

关键词: additive bases;    sumsets;    representation functions;    Erdos Turan conjecture;    Konig's lemma;   

DOI  :  10.1016/j.jnt.2003.12.012

来源: Elsevier

【 摘 要 】

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) is an element of A and a(1) less than or equal to a(2) less than or equal to ... less than or equal toa(h) . The infinite A is called a basis of order h if r(A) (n, h) greater than or equal to 1 for every nonnegative integer n. Erdos and Turan conjectured that lim sup(n)-->(r) r(A)(n, 2) = infinity for every basis A of order 2. This paper introduces a new class of additive bases and a general additive problern, 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. (C) 2004 Elsevier Inc. All rights reserved.

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