On the discrepancy of some generalized Kakutani's sequences of partitions

In this paper we study a class of generalized Kakutani’s sequences of partitions of [0,1], constructed by using the technique of successive refinements. Our main focus is to derive bounds for the discrepancy of these sequences. The approach that we use is based on a tree representation of the sequence of partitions which is precisely the parsing tree generated by Khodak’s coding algorithm. With the help of this technique we derive (partly up to a logarithmic factor) optimal upper bound in the so-called rational case. The upper bounds in the irrational case that we obtain are weaker, since they heavily depend on Diophantine approximation properties of a certain irrational number. Finally, we present an application of these results to a class of fractals.