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On additive complements


Authors: Jin-Hui Fang and Yong-Gao Chen
Journal: Proc. Amer. Math. Soc. 138 (2010), 1923-1927
MSC (2010): Primary 11B13, 11B34
DOI: https://doi.org/10.1090/S0002-9939-10-10205-6
Published electronically: February 5, 2010
MathSciNet review: 2596025
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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$. For additive complements $ A$ and $ B$, Sárközy and Szemerédi proved that if $ \limsup\limits_{x\rightarrow\infty}\frac{A(x)B(x)}{x}\le 1$, then $ A(x)B(x)-x\rightarrow+\infty$. In this paper, we prove that for additive complements $ A$ and $ B$, if $ \limsup\limits_{x\rightarrow\infty}\frac{A(x)B(x)}{x}<\frac 54$ or $ \limsup\limits_{x\rightarrow\infty}\frac{A(x)B(x)}{x}>2$, then $ A(x)B(x)-x\rightarrow+\infty$.


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Additional Information

Jin-Hui Fang
Affiliation: School of Mathematical Sciences, Nanjing Normal University, Nanjing 210046, People’s Republic of China
Email: fangjinhui1114@163.com

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

DOI: https://doi.org/10.1090/S0002-9939-10-10205-6
Keywords: Additive complements, sequences, counting functions
Received by editor(s): July 1, 2009
Received by editor(s) in revised form: September 10, 2009
Published electronically: February 5, 2010
Additional Notes: This work was supported by the National Natural Science Foundation of China, Grant No. 10771103 and the Outstanding Graduate Dissertation Program of Nanjing Normal University, Grant No. 181200000213.
Communicated by: Wen-Ching Winnie Li
Article copyright: © Copyright 2010 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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