On the distance sets of Ahlfors-David regular sets

Abstract

prove that if empty set not equal K subset of R-2 is a compact s-Ahlfors-David regular set with s >= 1, then dim(p) D(K) = 1, where D(K) := {vertical bar x - y vertical bar : x, y is an element of K} is the distance set of K, and dime stands for packing dimension. The same proof strategy applies to other problems of similar nature. For instance, one can show that if empty set not equal K subset of R-2 is a compact s-Ahlfors David regular set with s >= 1, then there exists a point x(0) is an element of K such that dime K . (K - x(0)) = 1. (C) 2016 Elsevier Inc. All rights reserved.