The relative growth of information in two-dimensional partitions.

Abstract

Let $\overline{x} \in [0,1)^2$. In this paper we find the rate at which knowledge about the partition elements $\overline{x}$ lies in for one sequence of partitions determines the partition elements it lies in for another sequence of partitions. This rate depends on the entropy of these partitions and the geometry of their shapes, and gives a two-dimensional version of Lochs' theorem.