  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.
