  Recent work of Kurasov and Sarnak provides a method for constructing
one-dimensional Fourier quasicrystals (FQ) from the torus zero sets of a
special class of multivariate polynomials called Lee-Yang polynomials. In
particular, they provided a non-periodic FQ with unit coefficients and
uniformly discrete support, answering an open question posed by Meyer. Their
method was later shown to generate all one-dimensional Fourier quasicrystals
with $\mathbb{N}$-valued coefficients ($ \mathbb{N} $-FQ).
  In this paper, we characterize which Lee-Yang polynomials give rise to
non-periodic $ \mathbb{N} $-FQs with unit coefficients and uniformly discrete
support, and show that this property is generic among Lee-Yang polynomials. We
also show that the infinite sequence of gaps between consecutive atoms of any
$\mathbb{N}$-FQ has a well-defined distribution, which, under mild conditions,
is absolutely continuous. This generalizes previously known results for the
spectra of quantum graphs to arbitrary $\mathbb{N}$-FQs. Two extreme examples
are presented: first, a sequence of $\mathbb{N}$-FQs whose gap distributions
converge to a Poisson distribution. Second, a sequence of random Lee-Yang
polynomials that results in random $\mathbb{N}$-FQs whose empirical gap
distributions converge to that of a random unitary matrix (CUE).
