Bounds for $L_p$-discrepancies of point distributions in compact metric measure spaces

It will be shown in the talk that nontrivial upper bounds for the $L_p$-discrepancies of point distributions in compact metric measure spaces can be proved for all exponents $0 < p < \infty$ and $p = \infty$ under very simple conditions on the volume of metric balls as a function of radii. Particularly, these conditions hold for all compact Riemannian manifolds. Such upper bounds are sharp, at least, for $2 \le p < \infty$ and Riemannian symmetric manifolds of rank one. (The paper with the detailed proofs is available at arXiv: 1802.01577).