On additive complements. II
On additive complements. II
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DOI：
10.1090/s0002-9939-2010-10652-4
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发表时间：
2011-03
期刊：
影响因子：
--
通讯作者：
Yong-Gao Chen;Jin-Hui Fang
中科院分区：
文献类型：
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作者：
Yong-Gao Chen;Jin-Hui Fang
两个非负整数的无限序列A和B称为加法补，如果它们的和包含所有足够大的整数。设A（x）和B（x）是A和B的计数函数，令lim sup A（x）B（x）/x = α（A，B）.最近，作者[Proceedings of the x→∞ American Mathematical Society 138（2010），1923-1927]证明了对加法补数A和B，若α（A，B）2，则当x → ∞时A（x）B（x）-x → +∞.本文证明了对任意e > 0，存在加法补数A和B，且2-e < α（A，B）< 2，且对无穷多个正整数x，A（x）B（x）-x = 1.
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 lim sup A(x)B(x)/x = α(A,B). Recently, the authors [Proceedings of the x→∞ American Mathematical Society 138 (2010), 1923-1927] proved that for additive complements A and B, if α(A, B) 2, then A(x)B(x) -x → +∞ as x → ∞. In this paper, we prove that for any e > 0 there exist additive complements A and B with 2-e < α(A, B) < 2 and A(x)B(x)-x = 1 for infinitely many positive integers x.