  A random set of points in Euclidean space is called `rigid' or `hyperuniform'
if the number of points falling inside any given region has significantly
smaller fluctuations than the corresponding number for a set of i.i.d. random
points. This phenomenon has received considerable attention in recent years,
due to its appearance in random matrix theory, the theory of Coulomb gases and
zeros of random analytic functions. However, most of the published results are
in dimensions one and two. This paper gives the first proof of hyperuniformity
in a Coulomb type system in dimension three, known as the hierarchical Coulomb
gas. This is a simplified version of the actual 3D Coulomb gas. The interaction
potential in this model, inspired by Dyson's hierarchical model of the Ising
ferromagnet, has a hierarchical structure and is locally an approximation of
the Coulomb potential. Hyperuniformity is proved at both macroscopic and
microscopic scales, with upper and lower bounds for the order of fluctuations
that match up to logarithmic factors. The fluctuations have cube-root behavior,
in agreement with a well-known prediction for the 3D Coulomb gas. For
completeness, analogous results are also proved for the 2D hierarchical Coulomb
gas and the 1D hierarchical log gas.
