  Hyperuniformity is the study of stationary point processes with a sub-Poisson
variance in a large window. In other words, counting the points of a
hyperuniform point process that fall in a given large region yields a
small-variance Monte Carlo estimation of the volume. Hyperuniform point
processes have received a lot of attention in statistical physics, both for the
investigation of natural organized structures and the synthesis of materials.
Unfortunately, rigorously proving that a point process is hyperuniform is
usually difficult. A common practice in statistical physics and chemistry is to
use a few samples to estimate a spectral measure called the structure factor.
Its decay around zero provides a diagnostic of hyperuniformity. Different
applied fields use however different estimators, and important algorithmic
choices proceed from each field's lore. This paper provides a systematic survey
and derivation of known or otherwise natural estimators of the structure
factor. We also leverage the consistency of these estimators to contribute the
first asymptotically valid statistical test of hyperuniformity. We benchmark
all estimators and hyperuniformity diagnostics on a set of examples. In an
effort to make investigations of the structure factor and hyperuniformity
systematic and reproducible, we further provide the Python toolbox
structure_factor, containing all the estimators and tools that we discuss.
