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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).
$\mathrm{M}/\mathrm{G}/\infty$ queues
We consider an 
$\mathrm{M}/\mathrm{G}/\infty$ queue with infinite expected service time. We then provide the transience/recurrence classification of the states (the system is said to be at state n if there are n customers being served), observing also that here (unlike irreducible Markov chains, for example) it is possible for recurrent and transient states to coexist. We also prove a lower bound on the growth speed in the transient case.
We consider two continuous-time generalizations of conservative random walks introduced in Englander and Volkov (2022), an orthogonal and a spherically symmetrical one; the latter model is also known as random flights. For both models, we show the transience of the walks when 
$d\ge 2$ and that the rate of direction changing follows a power law 
$t^{-\alpha}$, 
$0<\alpha\le 1$, or the law 
$(\!\ln t)^{-\beta}$ where 
$\beta>2$.
Let 
$(X,\mu ,T,d)$ be a metric measure-preserving dynamical system such that three-fold correlations decay exponentially for Lipschitz continuous observables. Given a sequence 
$(M_k)$ that converges to 
$0$ slowly enough, we obtain a strong dynamical Borel–Cantelli result for recurrence, that is, for 
$\mu $-almost every 
$x\in X$, where 
$$ \begin{align*} \lim_{n \to \infty}\frac{\sum_{k=1}^{n} \mathbf{1}_{B_k(x)}(T^{k}x)} {\sum_{k=1}^{n} \mu(B_k(x))} = 1, \end{align*} $$
$\mu (B_k(x)) = M_k$. In particular, we show that this result holds for Axiom A diffeomorphisms and equilibrium states under certain assumptions.
