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| Autores principales: | , |
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| Formato: | Preprint |
| Publicado: |
2024
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| Acceso en línea: | https://arxiv.org/abs/2410.22664 |
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| _version_ | 1866915726885912576 |
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| author | Patil, Bhuwanesh Rao Mohan |
| author_facet | Patil, Bhuwanesh Rao Mohan |
| contents | Let $A$ be a set of natural numbers. A set $B$, a set of natural numbers, is an additive complement of the set $A$ if all sufficiently large natural numbers can be represented in the form $x+y$, where $x\in A$ and $y\in B$. Erdős proposed a conjecture that every infinite set of natural numbers has a sparse additive complement, and in 1954, Lorentz proved this conjecture. This article describes the existence or non-existence of those additive complements of the set $A$ that is a subset of the complement of $A$. We provide a ratio test to verify the existence of such additive complements. In precise, we prove that if $A=\{a_i: i\in \mathbb{N}\}$ is a set of natural numbers such that $a_i<a_{i+1} $ for $i \in \mathbb{N}$ and $\liminf_{n\rightarrow \infty } (a_{n+1}/a_{n})>1$, then there exists a set $B\subset \mathbb{N}\setminus A$ such that $B$ is a sparse additive complement of the set $A$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2410_22664 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | On additive complements in the complement of a set of natural numbers Patil, Bhuwanesh Rao Mohan Number Theory Let $A$ be a set of natural numbers. A set $B$, a set of natural numbers, is an additive complement of the set $A$ if all sufficiently large natural numbers can be represented in the form $x+y$, where $x\in A$ and $y\in B$. Erdős proposed a conjecture that every infinite set of natural numbers has a sparse additive complement, and in 1954, Lorentz proved this conjecture. This article describes the existence or non-existence of those additive complements of the set $A$ that is a subset of the complement of $A$. We provide a ratio test to verify the existence of such additive complements. In precise, we prove that if $A=\{a_i: i\in \mathbb{N}\}$ is a set of natural numbers such that $a_i<a_{i+1} $ for $i \in \mathbb{N}$ and $\liminf_{n\rightarrow \infty } (a_{n+1}/a_{n})>1$, then there exists a set $B\subset \mathbb{N}\setminus A$ such that $B$ is a sparse additive complement of the set $A$. |
| title | On additive complements in the complement of a set of natural numbers |
| topic | Number Theory |
| url | https://arxiv.org/abs/2410.22664 |