A generalization of a theorem of Erdős on asymptotic basis of order 2
type: Item
instance of: scholarly article
title: A generalization of a theorem of Erdős on asymptotic basis of order 2
zbMATH Open document ID: 0812.11011
author: Martin Helm
published in: Journal de Théorie des Nombres de Bordeaux
publication date: 1995-05-15T00:00:00Z
full work available at URL: http://www.numdam.org/item
full work available at URL: https://eudml.org/doc/226347
full work available at URL: jtnb6-1.html
review text: A classical theorem of Erdős asserts the existence of a basis of order 2 such that the number of representations satisfies \(r(n) \asymp \log n\). Here the following analogue is proved. For every \(k\) there is a set \(A\) and a decomposition of the set of integers into disjoint sets \(T_ j\) such that the counting function of each \(T_ j\) is of order \(\log_{k- 1} n\), while \(r(n)\), the number of representations of \(n\) as a sum of two elements of \(A\), satisfies \(r(n) \asymp \log_ k n\) on each \(T_ j\), that is, for each \(j\) the set \[ \{n\in T_ j:\;r(n)\not\in (c_ 1\log_ k n, c_ 2\log_ k n)\} \] is finite. (\(\log_ k n\) is the \(k\) times iterated logarithm).
Mathematics Subject Classification ID: 11B13
Mathematics Subject Classification ID: 11B83
zbMATH DE Number: 703934
zbMATH Keywords: asymptotic basis of order 2
zbMATH Keywords: additive bases
reviewed by: Imre Z. Ruzsa
MaRDI profile type: Publication
OpenAlex ID: W2332447908
cites work: Q3234565
cites work: Additive properties of random sequences of positive integers
cites work: Q3963072
cites work: Q3825162
DOI: 10.5802/JTNB.103
description: scientific article; zbMATH DE number 703934