Similar point configurations via group actions

We prove that for d≥2, k≥2, if the Hausdorff dimension of a compact set E⊂Rd is greater than d22d−1, then, for any given r>0, there exist (x1,…,xk+1)∈Ek+1, (y1,…,yk+1)∈Ek+1, a rotation θ∈Od(R), and a vector a∈Rd such that rxj=θyj−a for 1≤j≤k+1. Such a result on existence of similar k-simplices in thin sets had previously been established under a more stringent dimensional threshold in Greenleaf et al. The argument we use to prove the main result here was previously employed by Bhowmik and Rakhmonov to establish a finite field version. We also show the existence of multi-similarities of arbitrary multiplicity in Rd, and show how to extend these results from similarities to arbitrary proper continuous maps, as well as explore a general group-theoretic formulation of this problem in vector spaces over finite fields.