On a Problem of Nathanson Related to Minimal Additive Complements

Chen, Yong-Gao; Yang, Quan-Hui

doi:10.1137/12087339x

Cited by 18 publications

(20 citation statements)

References 5 publications

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“…Theorem 1 (Chen, Yang [8]). If W ⊆ Z is bounded neither above nor below, then W has a minimal additive complement.…”

Section: Introductionmentioning

confidence: 99%

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Sets Arising as Minimal Additive Complements in the Integers

Burcroff¹,

Luntzlara²

2020

Preprint

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In this paper, we study which sets of integers arise as minimal additive complements. We confirm a conjecture of Kwon, showing that bounded-below sets with arbitrarily large gaps arise as minimal additive complements. Moreover, our construction shows that any such set belongs to a co-minimal pair, strengthening a result of Biswas and Saha for lacunary sequences. We bound the upper and lower Banach density of syndetic sets that arise as minimal additive complements to finite sets. We provide some necessary conditions for an eventually periodic set to arise as a minimal additive complement and demonstrate that these necessary conditions are also sufficient for certain classes of eventually periodic sets. We conclude with several conjectures and questions concerning the structure of minimal additive complements.

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“…Theorem 1 (Chen, Yang [8]). If W ⊆ Z is bounded neither above nor below, then W has a minimal additive complement.…”

Section: Introductionmentioning

confidence: 99%

“…Theorem 2 (Chen, Yang [8]). Let W = {w 1 < w 2 < • • • } be a set of integers and Kwon [14] initiated the study of the dual analogue to Nathanson's question, asking which sets of integers arise as minimal additive complements.…”

Section: Introductionmentioning

confidence: 99%

Sets Arising as Minimal Additive Complements in the Integers

Burcroff¹,

Luntzlara²

2020

Preprint

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“…If inf W = −∞ and sup W = ∞, then there exists a minimal additive complement to W . Theorem 3 (Chen, Yang [1]). Let W = {1 = w1 < w2 < · · · } be a set of integers and W = N \ W = {w1 < w2 < · · · }.…”

Section: Introductionmentioning

confidence: 99%

“…Theorem 4 (Kiss, Sándor, Yang [2]). Let W = (nN + A) ∪ F ∪ G be as in (1). If W has a minimal additive complement, then there exists S ⊆ Z/nZ such that the following two conditions hold:…”

Section: Introductionmentioning

confidence: 99%

A note on minimal additive complements of integers

Kwon¹

2019

Discrete Mathematics

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“…For the second part of the above problem, in 2012, Chen and Yang [2] gave two infinite sets W 1 and W 2 of integers such that there exists a complement to W 1 that does not contain a minimal complement and every complement to W 2 contains a minimal complement.…”

Section: Introductionmentioning

confidence: 99%

On minimal additive complements of integers

Kiss

Sándor

Yang

2019

Journal of Combinatorial Theory, Series A

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Let C, W ⊆ Z. If C + W = Z, then the set C is called an additive complement to W in Z. If no proper subset of C is an additive complement to W , then C is called a minimal additive complement. Let X ⊆ N. If there exists a positive integer T such that x+T ∈ X for all sufficiently large integers x ∈ X, then we call X eventually periodic.In this paper, we study the existence of a minimal complement to W when W is eventually periodic or not. This partially answers a problem of Nathanson.

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On a Problem of Nathanson Related to Minimal Additive Complements

Cited by 18 publications

References 5 publications

Sets Arising as Minimal Additive Complements in the Integers

Sets Arising as Minimal Additive Complements in the Integers

A note on minimal additive complements of integers

On minimal additive complements of integers

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