Generalized additive bases, König's lemma, and the Erdős–Turán conjecture
✍ Scribed by Melvyn B. Nathanson
- Publisher
- Elsevier Science
- Year
- 2004
- Tongue
- English
- Weight
- 211 KB
- Volume
- 106
- Category
- Article
- ISSN
- 0022-314X
No coin nor oath required. For personal study only.
✦ Synopsis
Let A be a set of nonnegative integers. For every nonnegative integer n and positive integer h; let r A ðn; hÞ denote the number of representations of n in the form n
where a 1 ; a 2 ; y; a h AA and a 1 pa 2 p?pa h : The infinite set A is called a basis of order h if r A ðn; hÞX1 for every nonnegative integer n: Erdo +s and Tura´n conjectured that lim sup n-N r A ðn; 2Þ ¼ N 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 Erdo +s-Tura´n conjecture. Ko¨nig'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.