On additive complements. III
type: Item
instance of: scholarly article
title: On additive complements. III
zbMATH Open document ID: 1370.11018
author: Yonggao Chen
author: Jinhui Fang
published in: Journal of Number Theory
publication date: 2014-08-28T00:00:00Z
reviewed by: Georges Grekos
Mathematics Subject Classification ID: 11B13
zbMATH DE Number: 6335260
zbMATH Keywords: additive complements
MaRDI profile type: Publication
full work available at URL: j.jnt.2014.01.027
OpenAlex ID: W2003711701
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DOI: 10.1016/J.JNT.2014.01.027
review text: Two sets \(A,B\) of non-negative integers are called ``additive complements'' if every sufficiently large positive integer is sum of two numbers, one of them belonging to \(A\) and the other one to \(B.\) Of course \(A\) and \(B\) may have elements in common or even be equal.NEWLINENEWLINEIt is known (see [\textit{J. Fang} and \textit{Y. Chen}, Proc. Am. Math. Soc. 138, No. 6, 1923--1927 (2010; Zbl 1202.11010)] for a reference) that the counting functions \(A(x)=|A\cap[1,x]|\) and \(B(x)\) of two additive complements \(A\) and \(B\) verify \(A(x)B(x)>x-c\) for a constant \(c\) and all \(x\geq 1\). Consequently NEWLINE\[NEWLINE\liminf_{x\to+\infty}x^{-1}A(x)B(x)\geq 1.NEWLINE\]NEWLINE \textit{A. Sárközy} and \textit{E. Szemerédi} [Acta Math. Hung. 64, No. 3, 237--245 (1994; Zbl 0816.11013)] proved in 1994 that ifNEWLINENEWLINE\noindent \(\limsup_{x\rightarrow+\infty}x^{-1}A(x)B(x)\leq1\) (that is, if \(\lim_{x\rightarrow+\infty}x^{-1}A(x)B(x)=1),\) then NEWLINE\[NEWLINE\lim_{x\to+\infty}A(x)B(x)-x=+\infty.\tag{1} NEWLINE\]NEWLINENEWLINENEWLINEThe authors of the paper under review proved in [Proc. Am. Math. Soc. 138, No. 6, 1923--1927 (2010; Zbl 1202.11010)] that (1) still holds when NEWLINE\[NEWLINE\limsup_{x\rightarrow+\infty}x^{-1}A(x)B(x)>2 ~~\text{or}~~\limsup_{x\rightarrow+\infty}x^{-1}A(x)B(x)<{5\over 4}.NEWLINE\]NEWLINE In the above result, the number 2 cannot be replaced by a smaller one: this was pointed out by the authors in [ibid. 139, No. 3, 881--883 (2011; Zbl 1202.11010)]. It was an open problem to replace \({5\over 4}\) by a bigger number. This is done in this paper: the authors managed to replace \({5\over 4}\) by \(3-\sqrt 3.\)NEWLINENEWLINENEWLINEThe paper is well written, contains a clear introduction to the subject and informs the reader about results on additive complements in the set of \textit{all} integers.NEWLINENEWLINENEWLINE\texttt{Open problems:} Is it possible to replace \(3-\sqrt 3\) by a bigger number? Possibly, find the biggest such nu…
description: scientific article; zbMATH DE number 6335260