Intersective sets for sparse sets of integers

Abstract

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).