Infinite unrestricted sumsets of the form $B+B$ in sets with large density
Fuente:
arXiv
Guardado en:
| Autores principales: | , |
|---|---|
| Formato: | Preprint |
| Publicado: |
2024
|
| Materias: | |
| Acceso en línea: | |
| Etiquetas: |
Agregar Etiqueta
Sin Etiquetas, Sea el primero en etiquetar este registro!
|
| _version_ | 1866929320248737792 |
|---|---|
| author | Kousek, Ioannis Radić, Tristán |
| author_facet | Kousek, Ioannis Radić, Tristán |
| contents | For a set $A \subset \mathbb{N}$ we characterize in terms of its density when there exists an infinite set $B \subset \mathbb{N}$ and $t \in \{0,1\}$ such that $B+B \subset A-t$, where $B+B : =\{b_1+b_2\colon b_1,b_2 \in B\}$. Specifically, when the lower density $\underline{d}(A) >1/2$ or the upper density $\overline{d}(A)> 3/4$, the existence of such a set $B\subset \mathbb{N}$ and $t\in \{0,1\}$ is assured. Furthermore, whenever $\underline{d}(A) > 3/4$ or $\overline{d}(A)>5/6$, we show that the shift $t$ is unnecessary and we also provide examples to show that these bounds are sharp. Finally, we construct a syndetic three-coloring of the natural numbers that does not contain a monochromatic $B+B+t$ for any infinite set $B \subset \mathbb{N}$ and number $t \in \mathbb{N}$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2404_12201 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Infinite unrestricted sumsets of the form $B+B$ in sets with large density Kousek, Ioannis Radić, Tristán Dynamical Systems Combinatorics Number Theory 37A05, 05D10, 11B13, 11B30 For a set $A \subset \mathbb{N}$ we characterize in terms of its density when there exists an infinite set $B \subset \mathbb{N}$ and $t \in \{0,1\}$ such that $B+B \subset A-t$, where $B+B : =\{b_1+b_2\colon b_1,b_2 \in B\}$. Specifically, when the lower density $\underline{d}(A) >1/2$ or the upper density $\overline{d}(A)> 3/4$, the existence of such a set $B\subset \mathbb{N}$ and $t\in \{0,1\}$ is assured. Furthermore, whenever $\underline{d}(A) > 3/4$ or $\overline{d}(A)>5/6$, we show that the shift $t$ is unnecessary and we also provide examples to show that these bounds are sharp. Finally, we construct a syndetic three-coloring of the natural numbers that does not contain a monochromatic $B+B+t$ for any infinite set $B \subset \mathbb{N}$ and number $t \in \mathbb{N}$. |
| title | Infinite unrestricted sumsets of the form $B+B$ in sets with large density |
| topic | Dynamical Systems Combinatorics Number Theory 37A05, 05D10, 11B13, 11B30 |
| url | https://arxiv.org/abs/2404.12201 |