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.
Date and time
Conference - DimaScat : Scattering Theory and Spectral Asymptotics of Differential Operators - in Honour of Dimitri Yafaev