Steklov Institute of Mathematics of the Russian Academy of Sciences, St. Petersburg

Point distributions in compact metric spaces

Abstract:

We consider finite point subsets (distributions) in compact metric spaces.
Our concern is with discrepancies of such distributions in metric balls and with
sums of distances between points of distributions. These characteristics are
not independent, for spheres they are related by the known Stolarsky's invariance
principle. It can be shown that a probabilistic version of this principle holds for very
general metric spaces.
It turns out that non-trivial upper bounds for discrepancies and lower bounds
for sums of distances can be also given for general rectifiable metric spaces.
Surprisingly enough, these bounds follows from quite elementary geometric and
combinatorial arguments.
At the same time, the proof of good lower bounds for discrepancies and upper bounds
for sums of distances is much more difficult and can be given for specific homogeneous
spaces such as compact Riemanian symmetric spaces of rank one. The proof is relaying
on spectral analysis on these spaces and involves detailed uniform asymptotic expansions
for the corresponding spherical functions.