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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

On additive complements. II


Authors: Yong-Gao Chen and Jin-Hui Fang
Journal: Proc. Amer. Math. Soc. 139 (2011), 881-883
MSC (2010): Primary 11B13, 11B34
Published electronically: September 29, 2010
MathSciNet review: 2745640
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Abstract: 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$ and let $ \limsup\limits_{x\rightarrow\infty}A(x)B(x)/ x$ $ =\alpha (A, B)$. Recently, the authors [Proceedings of the American Mathematical Society 138 (2010), 1923-1927] proved that for additive complements $ A$ and $ B$, if $ \alpha (A, B)<5/4$ or $ \alpha (A, B)>2$, then $ A(x)B(x)-x\rightarrow+\infty$ as $ x\to\infty $. In this paper, we prove that for any $ \varepsilon >0$ there exist additive complements $ A$ and $ B$ with $ 2-\varepsilon <\alpha (A, B) <2$ and $ A(x)B(x)-x=1$ for infinitely many positive integers $ x$.


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Additional Information

Yong-Gao Chen
Affiliation: School of Mathematical Sciences, Nanjing Normal University, Nanjing 210046, People’s Republic of China
Email: ygchen@njnu.edu.cn

Jin-Hui Fang
Affiliation: Department of Mathematics, Nanjing University of Information Science & Technology, Nanjing 210044, People’s Republic of China
Email: fangjinhui1114@163.com

DOI: http://dx.doi.org/10.1090/S0002-9939-2010-10652-4
PII: S 0002-9939(2010)10652-4
Keywords: Additive complements, sequences, counting functions.
Received by editor(s): April 14, 2010
Published electronically: September 29, 2010
Additional Notes: This work was supported by the National Natural Science Foundation of China, Grant No. 10771103.
Communicated by: Matthew A. Papanikolas
Article copyright: © Copyright 2010 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.