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Intersective sets for sparse sets of integers
- Part of
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- Journal:
- Ergodic Theory and Dynamical Systems , First View
- Published online by Cambridge University Press:
- 06 November 2024, pp. 1-33
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- Article
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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).
