Characterization of Entropy for Spacing shifts

Suppose $P\subseteq \mathbb{N}$ and let $(Σ_P,\,σ_P)$ be the space of a spacing shift. We show that if entropy $h_{σ_P}=0$ then $(Σ_P,\,σ_P)$ is proximal. Also $h_{σ_P}=0$ if and only if $P=\mathbb N\setminus E$ where $E$ is an intersective set. Moreover, we show that $h_{σ_P}>0$ implies that $P$ is a $Δ^*$ set; and by giving a class of examples, we show that this is not a sufficient condition. Then there is enough results to solve question 5 given in [J. Banks et al., \textit{Dynamics of Spacing Shifts}, Discrete Contin. Dyn. Syst., to appear.].

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