On additive complements. III

JOURNAL OF NUMBER THEORY

卷:141

Article

Fang, Jin-Hui¹  Chen, Yong-Gao^2,3 

[1] Nanjing Univ Informat Sci Technol, Dept Math, Nanjing 210044, Jiangsu, Peoples R China

[2] Nanjing Normal Univ, Sch Math Sci, Nanjing 210023, Jiangsu, Peoples R China

[3] Nanjing Normal Univ, Inst Math, Nanjing 210023, Jiangsu, Peoples R China

关键词: Additive complements;    Sequences;    Counting functions;   

DOI  :  10.1016/j.jnt.2014.01.027

来源: Elsevier

【 摘 要 】

Text. Two infinite sequences A and B of non-negative integers are called additive complements, if their sum contains all sufficiently large integers. Let A(x) and B(x) be the counting functions of A and B. In 1994, Sarkozy and Szemeredi proved that, for additive complements A and B, if lim sup A(x)B(x)/x <= 1, then A(x)B(x) - x -> +infinity as x -> +infinity. In 2010, the authors generalized this result and proved that if lim sup A(x)B(x)/x < 5/4 or lim sup A(x)B(x)/x > 2, then A(x)B(x) - x -> +infinity as x -> +infinity. In 2011, the authors pointed out that the constant 2 cannot be improved. In this paper, we improve 5/4 to 3 - root 3. (C) 2014 Elsevier Inc. All rights reserved.

【 授权许可】

Free