Pinned point configurations and Hausdorff dimension

Abstract

We prove that if the Hausdorff dimension of a compact subset E of Rd with d≥2 is sufficiently large, and if G is a star-like graph with two parts, and each of its parts is arigid 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 ifdimH(E)is sufficiently large, then∫νG(r~t)dνG(~t)>0,whereνGis 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.