Analysis Seminar

The classical Falconer distance problem asks for the threshold $s_0$ such that if the Hausdorff dimension of a compact subset of ${\Bbb R}^d$ is greater than $s_0$, then the set of distances determined by pairs of points of $E$ has positive Lebesgue measure. In this talk we are going to discuss a variant of this problem for configurations of many points. When the number of points does not exceed the ambient dimension plus one, the notion of congruence is still easily encoded by distances between the pairs of points. But when the number of points is much larger, interesting geometric phenomena come into play. We are also going to discuss some discrete variants of this problem that can be viewed as generalizations of the classical Erdos distance problem.