Intersective sets for sparse sets of integers
Fuente:
arXiv
Saved in:
| Main Authors: | , , , |
|---|---|
| Format: | Preprint |
| Published: |
2024
|
| Subjects: | |
| Online Access: | |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| _version_ | 1866911758435745792 |
|---|---|
| author | Bienvenu, Pierre-Yves Griesmer, John T. Le, Anh N. Lê, Thái Hoàng |
| author_facet | Bienvenu, Pierre-Yves Griesmer, John T. Le, Anh N. Lê, Thái Hoàng |
| contents | For $E \subset \mathbb{N}$, a subset $R \subset \mathbb{N}$ is $E$-intersective if for every $A \subset E$ having positive upper relative density, we have $R \cap (A - A) \neq \varnothing$. On the other hand, $R$ is chromatically $E$-intersective if for every finite partition $E=\bigcup_{i=1}^k E_i$, there exists $i$ such that $R\cap (E_i-E_i)\neq\varnothing$. When $E=\mathbb{N}$, we recover the usual notions of intersectivity and chromatic intersectivity.
In this article, we investigate to which extent known intersectivity results hold in the relative setting when $E = \mathbb{P}$, the set of primes, or other sparse subsets of $\mathbb{N}$. Among other things, we prove:
-There exists an intersective set that is not $\mathbb{P}$-intersective.
-However, every $\mathbb{P}$-intersective set is intersective.
-There exists a chromatically $\mathbb{P}$-intersective set which is not intersective (and therefore not $\mathbb{P}$-intersective).
-The set of shifted Chen primes $\mathbb{P}_{\mathrm{Chen}} + 1$ is $\mathbb{P}$-intersective (and therefore intersective). |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2401_07758 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Intersective sets for sparse sets of integers Bienvenu, Pierre-Yves Griesmer, John T. Le, Anh N. Lê, Thái Hoàng Number Theory Combinatorics Dynamical Systems For $E \subset \mathbb{N}$, a subset $R \subset \mathbb{N}$ is $E$-intersective if for every $A \subset E$ having positive upper relative density, we have $R \cap (A - A) \neq \varnothing$. On the other hand, $R$ is chromatically $E$-intersective if for every finite partition $E=\bigcup_{i=1}^k E_i$, there exists $i$ such that $R\cap (E_i-E_i)\neq\varnothing$. When $E=\mathbb{N}$, we recover the usual notions of intersectivity and chromatic intersectivity. In this article, we investigate to which extent known intersectivity results hold in the relative setting when $E = \mathbb{P}$, the set of primes, or other sparse subsets of $\mathbb{N}$. Among other things, we prove: -There exists an intersective set that is not $\mathbb{P}$-intersective. -However, every $\mathbb{P}$-intersective set is intersective. -There exists a chromatically $\mathbb{P}$-intersective set which is not intersective (and therefore not $\mathbb{P}$-intersective). -The set of shifted Chen primes $\mathbb{P}_{\mathrm{Chen}} + 1$ is $\mathbb{P}$-intersective (and therefore intersective). |
| title | Intersective sets for sparse sets of integers |
| topic | Number Theory Combinatorics Dynamical Systems |
| url | https://arxiv.org/abs/2401.07758 |