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 $A,B\subset\mathbb{R}^2$ have Hausdorff dimension $>1$ 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.