Pinned point configurations and Hausdorff dimension

Abstract

We prove that if the Hausdorff dimension of a compact subset E of R
d with
d ≥ 2 is sufficiently large, and if G is a star-like graph with two parts, and each of its parts is a
rigid graph, then the Lebesgue measure in the appropriate dimension, of the set of distances in
E specified by the graph is positive. We also prove that if dimH(E) is sufficiently large, then
Z
νG(r~t)dνG(~t) > 0,
where νG is the measure on the space of distances specified by G which is induced by a Frostman
measure. In particular, this means that for any r > 0 there exist many configurations encoded
by ~t with vertices in E such that the vertices of r~t are also in E .