  The notions of unimodular Minkowski and Hausdorff dimensions are defined in
[arXiv:1807.02980] for unimodular random discrete metric spaces. The present
paper is focused on the connections between these notions and the polynomial
growth rate of the underlying space. It is shown that bounding the dimension is
closely related to finding suitable equivariant weight functions (i.e.,
measures) on the underlying discrete space. The main results are unimodular
versions of the mass distribution principle, Billingsley's lemma and Frostman's
lemma, which allow one to derive upper bounds on the unimodular Hausdorff
dimension from the growth rate of suitable equivariant weight functions. These
results allow one to compute or bound both types of unimodular dimensions in a
large set of examples in the theory of point processes, unimodular random
graphs, and self-similarity. Further results of independent interest are also
presented, like a version of the max-flow min-cut theorem for unimodular
one-ended trees and a weak form of pointwise ergodic theorems for all
unimodular discrete spaces.
