  This paper addresses the problem of describing aperiodic discrete structures
that have a self-similar or self-affine structure. Substitution Delone set
families are families of Delone sets (X_1, ..., X_n) in R^d that satisfy an
inflation functional equation under the action of an expanding integer matrix
in R^d. This paper studies such functional equation in which each X_i is a
discrete multiset (a set whose elements are counted with a finite
multiplicity). It gives necessary conditions on the coefficients of the
functional equation for discrete solutions to exist. It treats the case where
the equation has Delone set solutions. Finally, it gives a large set of
examples showing limits to the results obtained.
