The problem of distributing a large number of points uniformly on a
compact set is an interesting and difficult problem which has
attracted much research.
In this talk, we will focus on recent work of the author, Grabner, Mullen
and Maymeskul which deals with numerical integration, Riesz configurations
and low discrepancy sequences on rectifiable sets. In particular,
we will study certain arrangements of $N\geq 1$ points on sets $A$
from a class $A^d$ of $d$-dimensional compact sets embedded in $R^d’$,
$1\leq d\leq d^’$. As an example, we can take the $d$ dimensional
unit sphere $S^d$ realized as a subset of $R^{d+1}$.
We assume that these points interact through a Riesz potential
$V=|\cdot|^{-s}$, where $s>0$
and $|\cdot|$ is the Euclidean distance in $R^d’$.
We will focus on the following ideas and new methods developed
in the case of the following items 1-3 below. Even in special cases such
as the sphere, most of what we develop below was until recently not known.
(1) The development of a numerical integration formula in terms of
Riesz energy which allows for discrepancy estimates on spheres for a large
class of smooth functions, typically Lipchitz of positive order.
(2) Estimates for separation and mesh norm of $0