  In this article we extend results of Kakutani, Adler--Flatto, Smilansky and
others on the classical $\alpha$-Kakutani equidistribution result for sequences
arising from finite partitions of the interval. In particular, we describe a
generalization of the equidistribution result to infinite partitions. In
addition, we give discrepancy estimates, extending results of Drmota--Infusino.
