Intersective sets for sparse sets of integers

Abstract

Abstract For $E \subset \mathbb {N}$ , a subset $R \subset \mathbb {N}$ is E-intersective if for every $A \subset E$ having positive relative density, $R \cap (A - A) \neq \varnothing $ . We say that 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. We investigate to what extent the 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 the following: (1) the set of shifted Chen primes $\mathbb {P}_{\mathrm {Chen}} + 1$ is both intersective and $\mathbb {P}$ -intersective; (2) there exists an intersective set that is not $\mathbb {P}$ -intersective; (3) every $\mathbb {P}$ -intersective set is intersective; (4) there exists a chromatically $\mathbb {P}$ -intersective set which is not intersective (and therefore not $\mathbb {P}$ -intersective).