On additive complements. II
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- by Yong-Gao Chen and Jin-Hui Fang PDF
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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 $x$.References
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Additional Information
- Yong-Gao Chen
- Affiliation: School of Mathematical Sciences, Nanjing Normal University, Nanjing 210046, People’s Republic of China
- MR Author ID: 304097
- Email: ygchen@njnu.edu.cn
- Jin-Hui Fang
- Affiliation: Department of Mathematics, Nanjing University of Information Science & Tech- nology, Nanjing 210044, People’s Republic of China
- Email: fangjinhui1114@163.com
- 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
- © Copyright 2010
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication. - Journal: Proc. Amer. Math. Soc. 139 (2011), 881-883
- MSC (2010): Primary 11B13, 11B34
- DOI: https://doi.org/10.1090/S0002-9939-2010-10652-4
- MathSciNet review: 2745640