On distance sets, box-counting and Ahlfors regular sets

Abstract

We obtain box-counting estimates for the pinned distance sets of (dense subsets of) planar discrete Ahlfors-regular sets of exponent s > 1. As a corollary, we improve upon a recent result of Orponen, by showing that if A is Ahlfors-regular of dimension s > 1, then almost all pinned distance sets of A have lower box-counting dimension 1. We also show that if A,B ⊂ R 2 have Hausdorff dimension greater than 1 and A is Ahlfors-regular, then the set of distances between A and B has modified lower box-counting dimension 1, which taking B = A improves Orponen’s result in a different direction, by lowering packing dimension to modified lower box-counting dimension. The proofs involve ergodic-theoretic ideas, relying on the theory of CP-processes and projections.