On additive complements

Fang, Jin-Hui; Chen, Yong-Gao

doi:10.1090/s0002-9939-10-10205-6

Cited by 14 publications

(5 citation statements)

References 5 publications

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“…This completes the proof of Lemma 2.1. Thus we could deduce from Theorem in [6] that A(x)B(x)−x → +∞ as x → +∞. By Lemma 2.1, we know that the above constants 2 and 4 cannot be improved.…”

Section: Proof Of Theorem 11mentioning

confidence: 91%

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On a property of additive complements

Fang¹,

Ma²

2020

Publ. Math. Debrecen

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Two infinite sequences A and B are called infinite additive complements if every sufficiently large integer can be expressed as the sum of two elements taken from A and B. Let A(x) (resp. B(x)) be the number of elements in A (resp. B) not exceeding x. Motivated by a recent result [3], the authors proved that, for infinite additive complements A, B, if lim sup A(2x)B(2x) A(x)B(x) < 2 or lim sup A(2x)B(2x) A(x)B(x) > 4, then A(x)B(x) − x → +∞ as x → +∞. Furthermore, the above constants 2 and 4 cannot be improved. we have A(x)B(x) − x → +∞ as x → +∞.

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Section: Proof Of Theorem 11mentioning

confidence: 91%

“…Chen and Fang (in [6] and [7] [1] showed that the constant 2 cannot be improved. Recently, Chen and Fang [3] further proved that, for two infinite sequences A and B, if there exists a constant 0 < c < 1 such that r(x) ≤ cx for all sufficiently large x and…”

Section: Introductionmentioning

confidence: 99%

On a property of additive complements

Fang¹,

Ma²

2020

Publ. Math. Debrecen

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“…In 1994, Sárközy and Szemerédi [2] proved the conjecture. Fang and Chen [3,4] On the other hand, Chen and Fang [5] proved the following result.…”

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confidence: 88%

A Note on Additive Complements

Ma¹

2022

Preprint

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“…See [Erd54], [Erd57] etc. Some more recent results on asymptotic complements can be found in [Wol96,FC10,CF11,FC14] etc.…”

Section: Introductionmentioning

confidence: 99%

Asymptotic complements in the integers

Biswas

Saha

2020

Journal of Number Theory

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Let W be a nonempty subset of the set of integers Z. A nonempty subset C of Z is said to be an asymptotic complement to W if W + C contains almost all the integers except a set of finite size. The set C is said to be a minimal asymptotic complement to W if C is an asymptotic complement to W , but C \ {c} is not an asymptotic complement to W for every c ∈ C. Asymptotic complements have been studied in the context of representations of integers since the time of Erdős, Hanani, Lorentz and others, while the notion of minimal asymptotic complements is due to Nathanson. In this article, we study minimal asymptotic complements in Z and deal with a problem of Nathanson on their existence and their inexistence.2010 Mathematics Subject Classification. 11B13, 11P70, 05E15, 05B10.

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On additive complements

Cited by 14 publications

References 5 publications

On a property of additive complements

On a property of additive complements

A Note on Additive Complements

Asymptotic complements in the integers

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