A Note on Minimal Additive Complements
Fuente:
arXiv
Salvato in:
| Autore principale: | |
|---|---|
| Natura: | Preprint |
| Pubblicazione: |
2017
|
| Soggetti: | |
| Accesso online: | |
| Tags: |
Aggiungi Tag
Nessun Tag, puoi essere il primo ad aggiungerne!!
|
| _version_ | 1866909368434294784 |
|---|---|
| author | Kwon, Andrew |
| author_facet | Kwon, Andrew |
| contents | Let $C, W \subseteq \mathbb{Z}$. If $C + W = \mathbb{Z}$, then the set $C$ is called an additive complement to $W$ in $\mathbb{Z}$. If no proper subset of $C$ is an additive complement to $W$, then $C$ is called a minimal additive complement. We provide a partial answer to a question posed by Kiss, Sándor, and Yang regarding the minimal additive complement of sets of the form $W = (n \mathbb{N} + A) \cup F \cup G$, where $|F|<\infty, (F \mod{n}) \subseteq (A \mod{n})$ and $(G \mod{n}) \cap (A \mod{n}) = \emptyset$. We also introduce the dual problem of characterizing sets that arise as the minimal additive complements of some set of integers, proving the analog of Nathanson's initial result on existence of minimal additive complements. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_1708_01287 |
| institution | arXiv |
| publishDate | 2017 |
| record_format | arxiv |
| spellingShingle | A Note on Minimal Additive Complements Kwon, Andrew Number Theory Combinatorics 11B13, 11B34 Let $C, W \subseteq \mathbb{Z}$. If $C + W = \mathbb{Z}$, then the set $C$ is called an additive complement to $W$ in $\mathbb{Z}$. If no proper subset of $C$ is an additive complement to $W$, then $C$ is called a minimal additive complement. We provide a partial answer to a question posed by Kiss, Sándor, and Yang regarding the minimal additive complement of sets of the form $W = (n \mathbb{N} + A) \cup F \cup G$, where $|F|<\infty, (F \mod{n}) \subseteq (A \mod{n})$ and $(G \mod{n}) \cap (A \mod{n}) = \emptyset$. We also introduce the dual problem of characterizing sets that arise as the minimal additive complements of some set of integers, proving the analog of Nathanson's initial result on existence of minimal additive complements. |
| title | A Note on Minimal Additive Complements |
| topic | Number Theory Combinatorics 11B13, 11B34 |
| url | https://arxiv.org/abs/1708.01287 |