Intersective sets for sparse sets of integers

Abstract

For E ⊂ N, a subset R ⊂ N is E-intersective if for every A ⊂ E having positive relative density, R ∩ (A − A) /= ∅. We say that R is chromatically E-intersective if for every finite partition E = ∪ki=1 Ei, there exists i such that R ∩ (Ei − Ei) /= ∅. When E = 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 = P, the set of primes, or other sparse subsets of N. Among other things, we prove the following: (1) the set of shifted Chen primes PChen + 1 is both intersective and P-intersective; (2) there exists an intersective set that is not P-intersective; (3) every P-intersective set is intersective; (4) there exists a chromatically P-intersective set which is not intersective (and therefore not P-intersective).