  City size distributions are known to be well approximated by power laws
across a wide range of countries. But such distributions are also meaningful at
other spatial scales, such as within certain regions of a country. Using data
from China, France, Germany, India, Japan, and the US, we first document that
large cities are significantly more spaced out than would be expected by chance
alone. We next construct spatial hierarchies for countries by first
partitioning geographic space using a given number of their largest cities as
cell centers, and then continuing this partitioning procedure within each cell
recursively. We find that city size distributions in different parts of these
spatial hierarchies exhibit power laws that are again far more similar than
would be expected by chance alone -- suggesting the existence of a spatial
fractal structure.
