Number rigid determinantal point processes induced by generalized Cantor sets

We consider the Ghosh-Peres number rigidity of translation-invariant determinantal point processes on the real line $\mathbb{R}$, whose correlation kernels are induced by the Fourier transform of the indicators of generalized Cantor sets in the unit interval. Our main results show that for any given $\theta \in (0,1)$, there exists a generalized Cantor set with Lebesgue measure $\theta$, such that the corresponding determinantal point process is Ghosh-Peres number rigid.