  Substitution schemes provide a classical method for constructing tilings of
Euclidean space. Allowing multiple scales in the scheme, we introduce a rich
family of sequences of tile partitions generated by the substitution rule,
which include the sequence of partitions of the unit interval considered by
Kakutani as a special case. Using our recent path counting results for directed
weighted graphs, we show that such sequences of partitions are uniformly
distributed, thus extending Kakutani's original result. Furthermore, we
describe certain limiting frequencies associated with sequences of partitions,
which relate to the distribution of tiles of a given type and the volume they
occupy.
